Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

This paper develops a macroscopic, activity-based model of urban active mobility using nonintrusive sensor data. It introduces attendance functions to describe spatio-temporal travel patterns between activities and formulates the disaggregation of aggregated counts as a statistical inference problem. Counts are modeled as Poisson variables, and unknown subpopulation sizes are estimated via maximum likelihood, with theoretical guarantees and an efficient EM algorithm for computation. Grounded in a microscopic stochastic model, the framework offers a scalable and privacy-preserving approach to analyzing urban soft mobility dynamics.

We study best-policy identification for finite-horizon risk-sensitive reinforcement learning under the entropic risk measure. Recent work established a constant gap in the exponential horizon dependence between lower and upper bounds on the number of samples required to identify an approximately optimal policy. Precisely, known lower bounds scale in $Ω(e^{|β| H})$ where $H$ is the horizon of the MDP, while the state-of-the-art upper bound achieves at best $O(e^{2|β| H})$ (arXiv:2506.00286v2) using a generative model. We show that this extra exponential factor can be traced to overly loose concentration control for exponential utilities. To close this open gap, we revisit the analysis of this problem through a forward-model based algorithm building on KL-based exploration bonuses that we adapt to the entropic criterion. The improvement we get is due to two main novel technical innovations. We leverage the smoothness properties of the exponential utility to derive sharper concentration bounds, and we propose a new stopping rule that exploits further this tightness to obtain a sample complexity that matches the lower bound.

In statistics permutations typically arise in the context of rank plots for two-dimensional data. Such plots can also be interpreted as discrete copulas. In discrete mathematics, typically in the context of the description of large (non-random) objects, two-dimensional copulas appear as limits of permutations and are then known as permutons if the topology refers to the convergence of pattern frequencies. We obtain a functional central limit theorem for such pattern frequencies in the context of two-dimensional random samples. The result serves as the basis for nonparametric goodness-of-fit tests, for two-sample tests, and for tests of symmetry. This includes a suitable variant of the bootstrap for obtaining critical values. Pattern-based procedures are also of interest in a parametric context. We consider two examples, the Farlie-Gumbel-Morgenstern class and a family of delay copulas. We discuss implementation aspects of the resulting procedures and we provide a simulation study that supplements the theoretical results in the nonparametric case.

With the growing application of spatial predictive modeling in ecology, the question of how to appropriately evaluate the resulting maps has gained increasing attention. While there is consensus that map accuracy is ideally estimated using an independent probability sample of the prediction area, there is still no agreement on the most appropriate way to conduct an evaluation for the common case when such a sample is not available. Cross-validation, which involves multiple train-test splits, is commonly applied not only to estimate final model accuracy but also to guide model tuning and selection. Many different spatial and non-spatial approaches to cross-validation have been proposed, and approaches in both groups have faced substantial criticism. It has been shown that random cross-validation methods are suitable when the training points are randomly distributed in the prediction area, while spatial cross-validation is better suited towards extrapolation situations. In practice, however, there is a continuum and most cases are between those two extremes. To address this gap, we advocate for a new category of cross-validation methods to account for this: prediction-domain adaptive evaluation. Methods in this category flexibly adapt to the prediction situation, yielding most reliable estimates of map accuracy across different scenarios. To ground this perspective empirically, we reproduce a simulation study that was used in earlier research and systematically compare different evaluation methods and discuss their purpose.

We introduce a family of synthetic languages with hierarchical structure -- generated by a broadcast process on trees -- for which the role of context length and reasoning in autoregressive generation can be analyzed precisely. At the heart of our analytic approach is an \emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the \emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the \emph{coloring broadcast process} (a hard-constrained language) in the freezing regime, bounded-context autoregression produces sequences that, with high probability, are inconsistent with \emph{any} valid coloring of the underlying tree. Together these results imply an $Ω(n)$ lower bound on the context length required to faithfully sample length-$n$ sequences. In contrast, we prove that an autoregressive \emph{reasoning} model with only $Θ(\log n)$ working memory can sample exactly from the true language -- an exponential improvement. We confirm both the lower-bound predictions and the reasoning-based upper bound empirically with transformers trained on the synthetic language; the trained models track our asymptotic predictions quantitatively across a wide range of context sizes.

Flow Language Models (FLMs) are a recently introduced class of language models which adapt continuous flow matching for one-hot encoded token sequences. Their denoisers have a special structure absent from generic continuous diffusion models: each block of the denoising mean is a posterior marginal distribution over the clean token at that position. Standard DDPM-style samplers collapse these marginals to a single conditional-mean endpoint and bridge toward this simplex-valued point, which is generally not a valid one-hot sequence. We argue that the natural sampler for an FLM is instead posterior-predictive. At each reverse step, we sample a clean one-hot endpoint from the factorized posterior defined by the FLM token marginals, and then sample the next continuous state from the analytic Ornstein--Uhlenbeck bridge conditioned on that endpoint. The method is training-free, uses the same model evaluations as standard sampling, and gives a principled interface for token-level decoding controls such as temperature scaling and nucleus truncation. We show that, under exact posterior marginals, the endpoint approximation error is exactly the conditional multi-information among token positions. The induced one-step bridge kernel preserves all token-wise posterior-predictive marginals and loses only the residual cross-position dependence. Finally, we prove a Girsanov path-space comparison showing that the marginal-conditioned bridge has a no-larger denoising-error term than the frozen conditional-mean bridge, with strict improvement whenever intermediate coordinate-wise bridge observations reveal additional information about the clean token. Experiments with FLMs show that the sampler improves the quality--diversity tradeoff. Code is available at: github.com/imbirik/mcb.

Camera traps have become a core tool in ecological research, enabling large-scale, noninvasive monitoring of wildlife populations and behavior. By automatically recording animals as they pass within view, these devices generate massive image datasets with minimal field effort. Yet this data richness introduces a new bottleneck when translating the images into usable information due to time and effort required for human annotation. Recently, artificial intelligent (AI) has been integrated into the workflow to improve this efficiency. However, the data procured from AI approaches are of a different nature, necessitating new statistical methods in order to obtain inference, make predictions, and quantify uncertainty. We propose a new Bayesian hierarchical data-fusion model which combines the strengths of human annotations and AI predictions. The benefits of our approach are an ability to provide uncertainty quantification as well as improved inference and prediction power, which we demonstrate using a simulation study. We apply our model to an AI analysis of the body condition of white-tailed deer (Odocoileus virginianus) from camera trap images from North Carolina to study the relationship between health and their environment. We find that bucks in rut have higher body condition than other deer and that green, open habitats are correlated with high body condition. Our new model derived novel ecological inference compared to a traditional approach using the same data.

Estimation of the extreme value index under right censoring is a fundamental problem in extreme value theory, with important applications in finance, insurance, and reliability. Classical integral estimators for Pareto-type tails typically require that the asymptotic proportion of uncensored observations in the tail is larger than one half, corresponding to the weak censoring regime. This restriction excludes many practically relevant situations involving strong censoring, where the proportion of uncensored observations is smaller than or equal to one half, and reflects the absence of a uniformly valid Gaussian approximation for the associated tail empirical process. To overcome this limitation, we introduce a weighted and truncated Nelson--Aalen tail empirical process and construct a class of integral estimators indexed by a tuning parameter larger than one. This approach restores a tractable asymptotic structure over the entire censoring range, from very weak to very strong censoring. Under standard regular variation conditions, we establish a uniform Gaussian approximation and derive consistency and asymptotic normality without imposing restrictions on the censoring level. A key ingredient of the analysis is a linearization of the estimator as a functional of the underlying process. Simulation studies and real data applications demonstrate improved stability and accuracy, particularly under moderate and strong censoring. In particular, the analysis of insurance loss data, representing weak censoring, and Australian AIDS survival data, representing strong censoring, illustrates the practical relevance of the proposed methodology across contrasting censoring regimes.

Most anomaly detection systems output scores rather than calibrated decisions, leaving practitioners to choose thresholds heuristically and without clear statistical interpretation. Conformal anomaly detection addresses this limitation by converting anomaly scores into calibrated p-values that are valid under the statistical assumption of data exchangeability, with a growing literature extending this idea beyond that setting. We present 'nonconform', a Python package for applying conformal anomaly detection within existing machine-learning workflows, and use it as the basis for an implementation-grounded introduction to the field. The package integrates with 'scikit-learn', 'pyod', and custom anomaly detectors, and provides a unified interface for calibration, p-value generation, and false discovery rate control. It supports several conformalization strategies, ranging from simple split-conformal calibration to more data-efficient and shift-aware extensions. Through a progression from foundational concepts to advanced conformalization strategies, complemented by code examples, the paper connects the statistical ideas behind conformal anomaly detection to their practical use in 'nonconform'. Empirical results demonstrate that the implemented methods enable statistically principled anomaly detection. Together, the package and exposition aim to make core conformal anomaly detection workflows more accessible and reproducible in experimental and production-oriented settings.

In this paper, we establish last-iterate convergence rates for off-policy actor--critic methods in reinforcement learning. In particular, under a single-loop, single-timescale implementation and a broad class of policy updates, including approximate policy iteration and natural policy gradient methods, we prove the first $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity guarantee for finding an $ε$-optimal policy under minimal assumptions, namely, the existence of a policy that induces an irreducible Markov chain. This stands in stark contrast to the existing literature, where an $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity is achieved only through nested-loop updates and/or under strong, algorithm-dependent assumptions on the policies, such as uniform mixing and uniform exploration. Technically, to address the challenges posed by the coupled update equations arising from the single-loop implementation, as well as the potentially unbounded iterates induced by off-policy learning, our analysis is based on a coupled Lyapunov drift framework. Specifically, we establish a geometric convergence rate for the actor and an $\tilde{\mathcal{O}}(1/T)$ convergence rate for the critic, and combine the two Lyapunov drift inequalities through a cross-domination property. We believe this analytical framework is of independent interest and may be applicable to other coupled iterative algorithms with unbounded

Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity,and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets.

ergodicity is an open-source Python library for computational work on stochastic dynamics, with particular emphasis on non-ergodicity, time-average behavior, heavy-tailed processes, and decision making under uncertainty. The package brings together three layers that are often split across ad hoc scripts: process definitions and simulators, analysis and fitting tools, and agent-based experimentation. This article documents the implemented software rather than presenting new stochastic theory. We describe the package architecture, the supported process families, the analysis workflow, and the practical boundaries of the current implementation. We also provide fully reproducible examples covering heavy-tailed ensemble spread, multiplicative Levy growth diagnostics, adaptive memory mean reversion, preasymptotic fluctuation analysis, and partial stochastic differential equation simulation. The package is positioned as an integration layer on top of the scientific Python stack, reducing the amount of glue code required to move from process specification to diagnostics and comparative experiments.

Causal discovery, the problem of inferring the direction of causality, is generally ill-posed. We use the language of structural causal models (SCM) to show that assuming that the causal relations are acyclic and invariant across multiple environments (e.g., the way minimum wage affects employment rate is stable across different geographical regions), \textit{only} two auxiliary environments are sufficient to infer the causal graph for arbitrary nonlinear mechanisms. Moreover, we demonstrate that this implies identifiability of the SCM functional mechanisms: as a corollary, we show that \textit{two} auxiliary environments are sufficient to guarantee correct counterfactual inference. We empirically support our theoretical results on synthetic data.

Causal discovery methods aim to infer causal direction from observational data. Functional causal discovery approaches use structural asymmetries to identify causal directionality but rely on strong modeling assumptions and provide limited tools for uncertainty quantification. We introduce Causal Discovery via Statistical Power (CDSP), a statistical inference framework that connects causal direction estimation with statistical power and enables uncertainty quantification. Considering the foundational setting of bivariate observational data, we show how quantities analogous to statistical power and effect size can be used in causal discovery to determine when data contain sufficient information to favor one direction over the other. We introduce the effect-size asymmetry assumption that characterizes when the probability of correctly detecting the causal direction (i.e., the power of causal discovery) exceeds that of incorrectly favoring the reverse direction. We show that the effect-size asymmetry assumption can be used for causal direction estimation with uncertainty quantification. Simulations show that CDSP direction estimation is robust to mild and moderate model misspecifications. Real data analyses on 100 cause-effect benchmark pairs further demonstrate that CDSP reduces false discovery rates by approximately 18% relative to a commonly used existing method.

The increasing availability of experimental data has intensified interest in calibrating stochastic models, raising fundamental questions about parameter identifiability. Structural identifiability determines whether parameters can be uniquely recovered from idealised, noise-free data, a prerequisite to allow for parameter estimation. However, existing methods to assess structural identifiability are not generally applicable to stochastic processes. We develop a methodology to analyse structural identifiability for a class of spatio-temporal stochastic processes. We investigate how identifiability depends on the type of available data, distinguishing between single-particle trajectories and total particle density measurements. For trajectory data, we use the individual-based model description that explicitly represents single-particle dynamics. For population-level data, we derive a partial differential equation model representation, that describes the evolution of total particle density, and apply a differential algebra approach, common to ordinary differential equations analysis. We further introduce a novel method to study the initial condition, based on characteristic equations to construct a Taylor expansion of the density evolution, enabling identification of additional identifiable parameter combinations. We apply our methodology to a model, and show it is identifiable with trajectory data but only locally identifiable with density data, and demonstrate the critical role of initial conditions in the identifiability analysis.

Calibration is commonly evaluated by comparing model confidence with its empirical correctness, implicitly treating reliability as a function of the confidence score alone. However, this view can hide substantial structure: models may be systematically overconfident on some kinds of inputs and underconfident on others, causing global reliability diagnostics to obscure localised calibration failures. To address this, we formulate the problem of discovering hidden miscalibration regimes without assuming access to predefined data slices. We define the corresponding miscalibration field and propose a diagnostic framework for estimating it. Our approach learns a calibration-aware representation of the input space and estimates signed local miscalibration by kernel smoothing in the learned geometry. Across four real-world LLM benchmarks and twelve LLMs, we find that input-dependent calibration heterogeneity is prevalent. We further show that the discovered fields are actionable: they support local confidence correction and reduce calibration error in systematically miscalibrated regions where confidence-based methods such as isotonic regression and temperature scaling are less effective.

Diffusion models are often trained in low-dimensional latent spaces, which are then reused for related but shifted datasets. In this work, we study when such latent reuse remains reliable under distribution shift. We consider a source-target setting in which both datasets are approximately low-dimensional but may lie near different subspaces. We show that freezing and reusing a source latent space induces a target-domain score error governed by two quantities: the principal-angle misalignment between the source and target subspaces, and the target ambient noise amplified by the diffusion time scale. Motivated by these limits, we further study mixed source-target training and characterize how the required shared latent dimension depends on the relative geometry of the two distributions. Our results provide theoretical guidance on when latent reuse is reliable and when learning a shared representation may be necessary.

Continuous intraday electricity markets play an increasingly important role in short-term trading and balancing, yet decision-making under rapidly evolving price dynamics remains challenging. This paper proposes a comprehensive framework for ensemble forecasting of intraday electricity price trajectories and their translation into adaptive trading decisions. Building on a corrected Support Vector Regression model, the approach extends point predictions to probabilistic trajectory forecasts by introducing scenario generation based on forecast errors of fundamental variables and proposing a novel Support Vector Sorting procedure for the efficient selection of representative scenarios. The framework is evaluated using transaction level data from the German intraday continuous market. Empirical results show improvements over benchmark methods in both statistical and economic terms. Fundamental scenarios enhance median trajectory accuracy but produce more concentrated predictive distributions, while historical simulation with scenario selection better captures tail risk. From an economic perspective, ensemble-based forecasts outperform naive benchmarks across most of the trading strategies. Dynamic updating through scenario reweighting further improves profitability with limited impact on downside risk. Overall, the results demonstrate that combining kernel-based learning with scenario driven uncertainty and adaptive updating provides a flexible and effective approach for forecasting and trading in continuous electricity markets.

A median-radius framework for assessing centrality in multivariate data using median distances is proposed. Based on the proposed framework, a scale invariant measure of radial dispersion is defined and used to establish a depth function that is robust to outliers and independent of covariance structure. The depth function does not depend on moment assumptions and naturally adapts to skewness, multimodality, and heavy-tailed distributions, which make it effective for high-dimensional data structures. We demonstrate fundamental characteristics of the underlying functionals such as subgradient and convexity. The subgradients provide additional insight and encode the imbalance in directional contributions of the data. This suggests a new approach to detect skewness and structural asymmetry through a purely radial construction. Empirical studies demonstrate that the method agrees with classical approaches under symmetry while providing a more flexible and informative characterization in complex settings.

Asynchronous stochastic gradient descent (ASGD) is a standard way to exploit heterogeneous compute resources in distributed learning: instead of forcing fast workers to wait for slow ones, the server updates the model whenever a gradient arrives. Vanilla ASGD applies each arriving gradient with the same weight. When local data distributions are heterogeneous, this becomes problematic: faster workers contribute more updates, and we show theoretically that the method is biased toward a frequency-weighted average of the local objectives rather than the desired global objective. Existing remedies typically move away from the simple ASGD template by introducing gathering phases, buffering, or extra memory. We show that this is unnecessary. Keeping the standard ASGD mechanism, we recover the correct objective by rescaling worker-specific stepsizes in proportion to their computation times, so that each worker contributes the same aggregate learning rate over a cycle. In the non-convex setting, under smoothness and bounded heterogeneity assumptions, we prove that the resulting method, Rescaled ASGD, converges to stationary points of the correct global objective in the fixed-computation model. Its time complexity matches the known lower bound in the leading term, while the effects of staleness and data heterogeneity appear only in lower-order terms. Experiments confirm that the method converges to the correct objective and is competitive with state-of-the-art baselines.

Many pre-trained models (PTMs) are available in modern applications. Because different PTMs are often trained on different datasets, their performances can vary substantially for different new tasks, and the ranking of the candidates may depend heavily on the input. Motivated by this, we propose a localized model averaging method with weights modeled as functions of the covariates, making it substantially more versatile than existing model averaging methods. This formulation allows the model averaging procedure to adaptively capture the varying relative advantages of different PTMs across heterogeneous contexts. Specifically, we learn flexible local weights under a general loss framework that accommodates a broad class of prediction tasks. We further establish the asymptotic optimality of the proposed method for both in-sample and out-of-sample risks, as well as the consistency of the estimated weights. Extensive numerical experiments further demonstrate the effectiveness of the proposed method.

We propose a data augmentation method for offline reinforcement learning, motivated by active positioning problems. Particularly, our approach enables the training of off-policy models from a limited number of suboptimal trajectories. We introduce a trajectory-based augmentation technique that exploits task structure and the geometric relationship between rewards, value functions, and mathematical properties of logging policies. During data collection, our augmentation supports suboptimal logging policies, leading to higher data quality and improved offline reinforcement learning performance. We provide theoretical justification for these strategies and validate them empirically across positioning tasks of varying dimensionality and under partial observability.

Applied researchers in biomedicine and related fields are often interested in estimating the causal effect of a treatment or intervention. Although randomized clinical trials are considered the gold standard for establishing causal effects, they are not always feasible, and real-world data may represent the only available source of evidence. In such settings, causal effects must be estimated using statistical methods applied to observational data. Over the last few decades, modern causal inference methods based on the potential outcomes framework have emerged as useful tools in this field. However, many such techniques exist, and their performance depends on factors such as sample size, the proportion of treated patients, the proportion of patients experiencing the outcome, the magnitude of the treatment effect, the target estimand, and potential violations of the fundamental assumptions of causal inference. Given the wide range of available methods, selecting an appropriate approach can be challenging for applied researchers. This study uses a large-scale simulation experiment to address this issue and provide researchers with a guide in the form of a handbook for a binary treatment and a binary outcome. Particularly, we test four popular statistical techniques: propensity score matching (full matching), inverse of the probability weighting, G-computation, and targeted maximum likelihood estimation. The proposed handbook is applied to two real-world datasets to assess its practical utility: one comprising vulnerable patients with mild COVID-19 (n=534 patients and more than 50% treated), and another of patients undergoing colorectal surgery (n=3635 patients and about 20% treated).

Generative models are often conditioned on a small set of examples via cross-attention. Under the Gaussian optimal-transport path, we show that the exact velocity field induced by a finite support set is a Nadaraya--Watson kernel smoother whose bandwidth decreases with flow time, from broad averaging at early steps to nearest-neighbor at late steps. A single Gaussian-kernel attention head exactly computes this field, connecting cross-attention conditioning to classical kernel theory. The theory predicts three failure regimes: nearest-neighbor collapse of the kernel at high dimension, mismatch between the isotropic kernel and the data geometry, and insufficient support for nonparametric estimation. Experiments on Gaussian mixtures, spherical shells, and DINOv2 ImageNet features confirm that learned conditioning improves in precisely these regimes, and that IP-Adapter's cross-attention implements approximate NW smoothing in practice.

The quantitative analysis of financial time series often reveals two distinct features that standard Gaussian frameworks fail to capture: heavy-tailed marginal distributions and the phenomenon of extreme co-movements.While extreme value theory characterizes marginal behavior, Copulas provide a functional bridge to describe the dependence structure independently of the marginals. We are proposing a different way of looking at the joint extremes on the basis of a dependence measure. The proposed idea incorporates both the non-identical and identical regularly varying distributions. Informed by the analysis of some high-frequency cryptocurrency datasets, the effect of persistence property have been thoroughly studied under these setups. A detailed simulation study confirms our intuition and findings.

In survival analysis, traditional models assume all individuals will eventually experience the event of interest. However, advances in therapeutics have led to multiple clinical contexts with potentially curative therapies, and in these contexts, certain individuals may never experience the event. Statisticians have developed cure models as a methodology to address this challenge. Nonetheless, despite significant statistical advances in cure models, we have seen more limited uptake in biomedical applications, and we hypothesize that this is caused by limited guidance in the appropriate application of cure models. Cure models require specific identifiability conditions for valid parameter estimation, and previous reports have demonstrated significant issues with the inappropriate application of cure models. Existing tutorials for cure models focus on model implementation and either assume or provide only limited guidance on whether cure modeling is appropriate for the given dataset. This tutorial addresses this gap by describing a systematic procedure that integrates clinical judgment, visual inspection of Kaplan-Meier curves, and quantitative evaluation. We provide a worked example using data from a randomized clinical trial in acute myeloid leukemia, and we also summarize findings from a series of other datasets of hematopoietic cell transplantation to suggest broad practical guidance for choosing to apply cure models. By systematically evaluating cure model appropriateness before fitting these models, researchers can achieve more reliable survival analysis and improved clinical decision-making.

The duality $L^{\infty}\simeq (L^{1})'$ frequently breaks down in the presence of model uncertainty, where a single reference measure $P$ is replaced by a non-dominated family of probability measures $\mathcal{P}$. The unavailability of classical measure-theoretic and functional-analytic tools in this regime poses a significant obstacle to developing robust probabilistic frameworks. We show that this duality can be restored for a broad class of robust statistical models by extending the underlying probability space. Specifically, on the extended model, the space $\mathbb{L}^{\infty}(\mathcal{P})$ of $\mathcal{P}$-quasi-surely bounded functions is isometrically isomorphic to the dual of the space of finite signed measures absolutely continuous with respect to at least one element of $\mathcal{P}$. The proposed extension is canonical: it is the smallest $\mathcal{P}$-complete extension of the original $σ$-algebra for which $\mathbb{L}^{\infty}(\mathcal{P})$ is the dual of any normed space. Our assumptions encompass several prominent non-dominated settings, including infinite product measures, Gaussian processes, the Black-Scholes model with uncertain constant volatility and drift, robust binomial models, and, more generally, infinite sequences from any parametric model with almost surely estimable parameters. Furthermore, we unify the existing frameworks of Cohen (2012) and Liebrich et al. (2022), demonstrating that our construction is equivalent to the capacity-based approach under mild assumptions satisfied by the aforementioned examples. Finally, we apply our theory to extend Kraft's (1955) characterization of strictly unbiased hypothesis tests to non-dominated cases.

Causal inference in continuous-time sequential decision problems is challenged by hidden confounders. We show that, in latent state-space models with time-varying interventions, observability of the latent dynamics from observed data is necessary for identifying dynamic treatment effects, linking control-theoretic observability to causal identifiability, even when hidden confounders affect both treatments and outcomes. We derive a continuous-time adjustment formula expressing potential outcome distributions under treatment trajectories via the measurement model, latent dynamics, and the filtering distribution over latent states given observed histories. We propose Observable Neural ODEs (ObsNODEs), Neural ODE models in observable normal form for causal forecasting. ObsNODEs learn continuous-time dynamics with states reconstructible from observations, enabling outcome prediction under alternative treatment paths. Experiments on synthetic cancer data, semi-synthetic data based on MIMIC-IV, and real-world sepsis data show strong performance over recent sequence models.

We consider predictive checking for Bayesian model assessment using leave-one-out probability integral transform (LOO-PIT). LOO-PIT values are conditional cumulative predictive probabilities given LOO predictive distributions and corresponding left out observations. For a well-calibrated model, LOO-PIT values should be near uniformly distributed, but in the finite sample case they are not independent, due to LOO predictive distributions being determined by nearly the same data (all but one observation). We prove that this dependency is non-negligible in the finite case and depends on model complexity. We propose three testing procedures that can be used for continuous and discrete dependent uniform values. We also propose an automated graphical method for visualizing local departures from the null. Extensive numerical experiments on simulated and real datasets demonstrate that the proposed tests achieve competitive performance overall and have much higher power than standard uniformity tests based on the independence assumption that inevitably lead to lower than expected rejection rate.

The concept of ranking aggregation plays a central role in preference analysis, and numerous algorithms for calculating median rankings, often originating in social choice theory, have been documented in the literature, offering theoretical guarantees in a centralized setting, \textit{i.e.}, when all the ranking data to be aggregated can be brought together in a single computing unit. For many technologies (\textit{e.g.} peer-to-peer networks, IoT, multi-agent systems), extending the ability to calculate consensus rankings with guarantees of convergence and resilience to potential contamination in a decentralized setting, when preference data is initially distributed across a communicating network, remains a major methodological challenge. Indeed, in recent years, the literature on decentralized computation has mainly focused on computing or optimizing statistics such as arithmetic means using gossip algorithms. The purpose of this article is precisely to study how to achieve reliable and resilient consensus on collective rankings in a decentralized setting, thereby raising new questions, robustness to corrupted nodes, and scalability through reduced communication costs in particular. The approach proposed and analyzed here relies on the robustness guarantees offered by random gossip communication, which allows autonomous agents to compute a global ranking consensus using local interactions only, without coordination or a central authority.

We formalize a generalized form of the Pareto principle - ``fraction $p$ of inputs yields fraction $1-p$ of outputs'' - as a property of non-negative gain densities $\ell \in L^1([0,1])$, working with the decreasing rearrangement to obtain a unique characterization. For probability distributions, the resulting $p$ coincides with $1 - k_F$, where $k_F$ is the Kolkata index of the corresponding Lorenz curve. Within this framework we analyze both constructed gain densities and commonly encountered distribution families. We derive closed-form expressions for $p$ for truncated power-law, exponential, and normal distribution families. Combining these with estimates of the truncation parameter as a function of sample size $N$, we predict that datasets of size $N \in [10^2, 10^5]$ from exponential and normal families concentrate $p$ near $[0.15, 0.26]$ and $[0.20, 0.29]$ - values close to the canonical 0.2/0.8-rule, and strictly below the saturation $k \approx 0.865$ conjectured earlier by Ghosh and Chakrabarti. We discuss the implications of the structural ubiquity of Pareto-type imbalances for their use as prescriptive targets.

We study a data-dependent notion of diffusion-model generalization: when a model does not memorize the training set, where do its generated samples go relative to the geometry induced by the data? To answer this, we introduce a time-dependent family of log-density ridge manifolds constructed from the smoothed empirical distribution, and use it to characterize reverse-time inference. Our main result shows that generated samples evolve by a reach-align-slide mechanism: they first enter a neighborhood of the ridge, then their distance to the ridge is controlled by the normal component of training error, and finally their motion along the ridge is controlled by the tangential component. We further connect this geometric picture to training dynamics through directional decompositions of the learned error, and make this link explicit for random feature models, where architectural bias and optimization error can be separated quantitatively. Experiments on synthetic multimodal data and MNIST latent diffusion support the predicted geometric behavior in both low and high dimensions.

We study supervised multiclass classification for diffusion processes, where each class is characterized by a distinct drift function and trajectories are observed at discrete times. We first derive a multidimensional Bayes rule and then construct a plug-in classifier by estimating the class-specific drifts with neural networks. Under standard regularity assumptions, we establish convergence rates for the excess misclassification risk, making explicit the contributions of drift estimation, time discretization, and dimension. Our analysis also highlights the benefit of exploiting the diffusion structure: the drift is learned from all observed increments, leading to sharper guarantees than direct trajectory-based neural classifiers in the considered setting. Numerical experiments support the theory: the proposed method achieves better classification performance than Denis et al. (2024) in dimension one, remains effective in higher dimensions when the drift functions admit a compositional structure, and outperforms end-to-end neural classifiers trained directly on trajectories, as in Bos & Schmidt-Hieber (2022).

The concept of independence plays a crucial role in probability theory and has been the subject of extensive research in recent years. Numerous approaches have been proposed to test for independence; however, most of them address the problem only at a global level. From a practical perspective, it is important not only to determine whether the data are dependent but also to identify where this dependence occurs and how strong it is. The graphical presentation of results is another essential aspect that should not be neglected, as it considerably enhances interpretability. The main objective of this work is to propose a solution that considers these aspects simultaneously. Relying on copula-based results, we introduce a novel method for testing global and local statistical independence using the quantile dependence function. Rather than assessing whether the value of the test statistic exceeds a single critical threshold and subsequently deciding whether to reject the independence hypothesis, we introduce so-called critical surfaces that guaranty a locally equal probability of exceeding them under independence. This approach enables a detailed examination of local discrepancies and an assessment of their statistical significance while preserving the overall significance level of the test.

An algorithm is proposed, analyzed, and tested for solving continuous nonlinear-equality-constrained optimization problems where the objective and constraint functions are defined by expectations or averages over large, finite numbers of terms. The main idea of the algorithm is to solve a sequence of related problems, each involving finite samples of objective- and constraint-function terms, over which the sample sets grow progressively. Under assumptions about the problem functions and their first- and second-order derivatives that are reasonable in real-world settings of interest, it is shown that -- with sufficiently large initial sample sizes -- solving a sequence of problems defined through progressive sampling yields a better worst-case sample complexity bound compared to solving a single problem with the full sets of samples. The results of numerical experiments with a set of test problems demonstrate that the proposed approach can be effective in practice.

Modern generative AI models, such as diffusion and flow matching models, can sample from rich data distributions. However, many applications, especially in science and engineering, require more than drawing samples from the model distribution: they require searching within this distribution for samples that optimise task-specific criteria. In this work, we propose O3 (Optimisation Over the Outputs of Generative Models), a method for sample-efficient black-box optimisation over continuous-variable diffusion and flow-matching models. O3 is built around surrogate latent spaces: low-dimensional Euclidean embeddings that can be extracted from a generative model without additional training. The resulting representations have controllable dimensionality and support the direct application of standard optimisation algorithms. We show, on image and protein design tasks, that surrogate-space optimisation finds substantially higher-scoring samples than standard sampling or optimisation in the original latent space. Our method is model- and optimiser-agnostic, incurs negligible additional cost over standard generation, and requires no retraining or fine-tuning of the generative model.

Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture-agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynolds-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.

This paper addresses the statistical estimation of Gaussian Mixture Models (GMMs) with unknown diagonal covariances from independent and identically distributed samples. We employ the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures, to simultaneously estimate the number of components and their parameters. Our main contribution extends the BLASSO methodology to multivariate GMMs with component-specific unknown diagonal covariance matrices. This setting is significantly more flexible than previous approaches, which required known and identical covariances. We establish non-asymptotic recovery guarantees with nearly parametric convergence rates for component means, diagonal covariances, and weights, as well as for density prediction. A key theoretical contribution is the identification of an explicit separation condition on mixture components that enables the construction of non-degenerate dual certificates-essential tools for establishing statistical guarantees for the BLASSO. Our analysis leverages the Fisher-Rao geometry of the statistical model and introduces a novel semi-distance adapted to our framework, providing new insights into the interplay between component separation, parameter space geometry, and achievable statistical recovery.

High-frequency mortality data have attracted growing attention, but their use has largely been confined to specific applications rather than general modelling and forecasting. Such data pose new challenges to traditional mortality models due to pronounced seasonal patterns and short-term fluctuations. To address these challenges and produce more accurate forecasts with the high-frequency mortality data, this paper introduces a novel integration of gradient boosting techniques into traditional stochastic mortality models under a multi-population setting. Our key innovation lies in using the Li and Lee model as the weak learner within the gradient boosting framework, replacing conventional decision trees. Empirical studies are conducted using weekly mortality data from 30 countries (Human Mortality Database, 2015-2019). Empirical evidence highlights that the proposed methodology not only enhances model fit by accurately capturing underlying mortality trends and seasonal patterns, but also achieves superior forecast accuracy, compared to the benchmark models. We also investigate a key challenge in multi-population mortality modelling: how to select appropriate sub-populations with sufficiently similar mortality experiences. A comprehensive clustering exercise is conducted based on mortality improvement rates and seasonal strength. The empirical results demonstrate that our proposed model maintains strong forecast accuracy across different clustering configurations, thereby reducing the need for extensive data preprocessing.

The Distributional Principal Autoencoder (DPA) combines distributionally correct reconstruction with principal-component-like interpretability of the encodings. In this work, we provide exact theoretical guarantees on both fronts. First, we derive a closed-form relation linking each optimal level-set geometry to the data-distribution score. This result explains DPA's empirical ability to disentangle factors of variation of the data, as well as allows the score to be recovered directly from samples. When the data follows the Boltzmann distribution, we demonstrate that this relation yields an approximation of the minimum free-energy path for the Mueller-Brown potential in a single fit. Second, we prove that if the data lies on a manifold that can be approximated by the encoder, latent components beyond the manifold dimension are conditionally independent of the data distribution - carrying no additional information - and thus reveal the intrinsic dimension. Together, these results show that a single model can learn the data distribution and its intrinsic dimension with exact guarantees simultaneously, unifying two longstanding goals of unsupervised learning.

We study the mixing time of a Susceptible--Infected--Susceptible (SIS) model on graphs with external sources of infection, which we refer to as the noisy SIS model. Under suitable assumptions on the parameters of the dynamics, we show that the mixing time is of the order $Θ(n\log n)$ with respect to the number of vertices $n$. We further investigate the model on random graph families, including Erd{ö}s--R{é}nyi graphs, random regular multigraphs, and Galton--Watson trees. By identifying high-probability structural properties of these graphs and conditioning on typical realizations, we prove that the mixing time remains of order $Θ(n\log n)$ with high probability.

Data scarcity poses a serious threat to modern machine learning and artificial intelligence, as their practical success typically relies on the availability of big datasets. One effective strategy to mitigate the issue of insufficient data is to first harness information from other data sources possessing certain similarities in the study design stage, and then employ the multi-task or meta learning framework in the analysis stage. In this paper, we focus on multi-task (or multi-source) linear models whose coefficients across tasks share an invariant low-rank component, a popular structural assumption considered in the recent multi-task or meta learning literature. Under this assumption, we propose a new algorithm, called Meta Subspace Pursuit (abbreviated as Meta-SP), that provably learns this invariant subspace shared by different tasks. Under this stylized setup for multi-task or meta learning, we establish both the algorithmic and statistical guarantees of the proposed method. Extensive numerical experiments are conducted, comparing Meta-SP against several competing methods, including popular, off-the-shelf model-agnostic meta learning algorithms such as ANIL. These experiments demonstrate that Meta-SP achieves superior performance over the competing methods in various aspects.

We give an order-explicit large deviation bound for the difference between a high-dimensional $U$-statistic and its Hájek projection. In particular, we show that any $U$-statistic of order $b$ on $n$ observations, with a $d$-dimensional kernel whose coordinates have $ψ_1$-Orlicz norm at most $ϕ$, has a maximum deviation from its Hájek projection of order $O_p(ϕb n^{-1}\log^2(dn))$. The proof relies on the development of novel order-explicit moment inequalities for higher-order Hoeffding components. We show that this rate is unimprovable, up to the polynomial factor on the logarithmic term. As corollaries, we obtain new Bernstein-type concentration and Gaussian approximation results for high-dimensional $U$-statistics. We apply these results to establish the consistency of a set of resampling-based simultaneous confidence intervals built around a class of nonparametric regression estimators constructed with subsampled kernels. This class encompasses several forms of random forest regression, including Generalized Random Forests.

Low-Rank Adaptation (LoRA) is the dominant parameter-efficient fine-tuning method due to its favorable compute-performance trade-off, yet it suffers from catastrophic forgetting. We study forgetting through a tractable _mean-field self-attention_ toy model, where tokens evolve as an interacting particle system and LoRA acts as a low-rank perturbation. Using tools from partial differential equations and dynamical systems, we characterize regimes suggesting a phase transition between forgetting and non-forgetting behavior. We show that one phase transition appears with respect to the norm of the perturbation, and the other with respect to the depth of the Transformers. We further bound the time-to-deviation in terms of the perturbation size and spectral quantities, and corroborate the predicted trends with experiments and exploratory analyses on real models under LoRA fine-tuning.

We show, using three empirical applications, that linear regression estimates predicated on the assumption of sparsity are fragile in two ways. First, we document that different choices of the regressor matrix which do not impact ordinary least squares (OLS) estimates, such as the choice of baseline category with categorical controls, can move sparsity-based estimates by two standard errors or more. Second, we develop two tests of the sparsity assumption by comparing sparsity-based estimators with OLS. The tests tend to reject the sparsity assumption in all three applications. Unless the number of regressors is comparable to or exceeds the sample size, OLS yields more robust inference at little efficiency cost.

Variational auto-encoders (VAEs) are a popular and powerful deep generative model. Previous works on VAEs have assumed a factorized likelihood model, whereby the output uncertainty of each pixel is assumed to be independent. This approximation is clearly limited as demonstrated by observing a residual image from a VAE reconstruction, which often possess a high level of structure. This paper demonstrates a novel scheme to incorporate a structured Gaussian likelihood prediction network within the VAE that allows the residual correlations to be modeled. Our novel architecture, with minimal increase in complexity, incorporates the covariance matrix prediction within the VAE. We also propose a new mechanism for allowing structured uncertainty on color images. Furthermore, we provide a scheme for effectively training this model, and include some suggestions for improving performance in terms of efficiency or modeling longer range correlations.

This paper is the first work to propose a network to predict a structured uncertainty distribution for a synthesized image. Previous approaches have been mostly limited to predicting diagonal covariance matrices. Our novel model learns to predict a full Gaussian covariance matrix for each reconstruction, which permits efficient sampling and likelihood evaluation. We demonstrate that our model can accurately reconstruct ground truth correlated residual distributions for synthetic datasets and generate plausible high frequency samples for real face images. We also illustrate the use of these predicted covariances for structure preserving image denoising.

This note addresses a key limitation of the Folding Test of Unimodality (FTU). In specific univariate mixture settings, the folding-based criterion can systematically fail, misclassifying clearly multimodal distributions as unimodal. We fully characterize these failures for Dirac mixtures and extend the analysis to Gaussian mixtures. We then introduce a double-folding procedure that captures complementary information, leading to a new test, the Double Folding Test of Unimodality. It resolves the FTU failures and improves multimodality detection power in simulations.

Most exploration algorithms search broadly until uncertainty is resolved. When the action space is too large to resolve within budget, practitioners default to $\varepsilon$-greedy, which bounds disruption but spends its override blindly. We introduce \textit{Delight-gated exploration} (DE), a host--override rule that spends exploratory actions only when their prospective delight (expected improvement times surprisal) exceeds a gate price. This practical heuristic recovers a classical result: Pandora's reservation-value rule for costly search, with surprisal setting the effective inspection cost. Resolved arms exit the gate, fresh arms shut off above a prior-determined threshold, and selected linear-bandit overrides consume finite information budget. Across Bernoulli bandits, linear bandits, and tabular MDPs, the same hyperparameters transfer without retuning, and DE shows much weaker regret growth than Thompson Sampling and $\varepsilon$-greedy in the tested unresolved regimes. Delight improves acting for the same reason it improves learning: it prices scarce resources by the product of upside and surprisal.

Recent advancements in large language models demonstrate that injecting perturbations can substantially enhance extrapolation performance. However, current approaches often rely on discrete perturbations with fixed designs, which limits their flexibility. In this work, we propose a framework where token prefixes are perturbed by a learnable transformation of a continuous latent vector within an embedding space. To overcome the challenge of an intractable marginal likelihood, we derive unbiased estimating equations for model parameters and optimize them via stochastic gradient descent. We establish the statistical properties of the resulting estimator in over-parameterized regimes. Empirical evaluations on both synthetic and real-world datasets demonstrate that our proposal yields significant gains in out-of-domain settings over a range of state-of-the-art baseline methods.

We revisit Byzantine robust distributed estimation for high-dimensional sparse linear models. By combining local $\ell_1$-regularized robust estimation with robust aggregation at the server, the framework applies to pseudo-Huber regression, quantile regression, and sparse SVM. We show that the resulting estimators yield non-asymptotic guarantees and attain near-optimal statistical rates under mild conditions, while remaining communication-efficient. Simulations confirm strong robustness in estimation, support recovery and classification accuracy under various Byzantine attacks.

Physics-Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high-dimensional space, enabling the use of Koopman-based analysis and showing that high-rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.

We study multiple change point localization under bandit feedback. An unknown piecewise-constant function on a compact interval can be queried sequentially at adaptively chosen inputs, and each query returns a noisy evaluation of the function. The goal is to identify a prescribed number of discontinuities, known as change points, within a target precision $η$ and confidence level $1-δ$, while using as few samples as possible. We propose an adaptive algorithm that first detects intervals likely to contain change points and then refines their locations to precision $η$. We establish non-asymptotic upper bounds on its sample budget, together with corresponding lower bounds. Prior work shows that jump magnitudes alone determine the asymptotic sample complexity as $δ\to 0$. We reveal that this picture is incomplete beyond this regime. We demonstrate, both empirically and theoretically, that for general $δ$ and $η$, the complexity is jointly governed by the jumps and the relative positions of the change points.

Large language models (LLMs) are increasingly deployed in settings where the available context is incomplete or degraded. We argue that an LLM generating answers under incomplete context can be viewed as an implicit imputer, and evaluated against a criterion from the multiple imputation (MI) literature: uncertainty should scale with the amount of missing information. We assess this criterion on SQuAD, using a controlled framework in which context availability is varied across five levels. We evaluate two answer-level uncertainty measures that can be estimated from repeated sampling: sampling-based confidence (empirical mode frequency) and response entropy. Confidence fails to reflect increasing missingness: it remains high even as accuracy collapses. Entropy, by contrast, increases with context removal, consistent with the MI analogy, and explains substantially more variance in accuracy than confidence across all evidence levels (quadratic $R^2$ gap up to 0.057). We further introduce a black-box diagnostic $ρ_R(α)$ that estimates the proportion of baseline uncertainty resolved by context level $α$, requiring only repeated sampling with and without context. These results suggest that entropy is a more responsive black-box uncertainty measure than confidence under incomplete context.

This work proposes $χ^2$-type test statistics to assess different hypotheses on the local structure of an observed marked point pattern. The test statistics is based on the local inhomogeneous extension of the mark-weighted $K$-function to investigate local behaviour of the marked point pattern. The summary statistic captures interactions between marks and locations by assessing local contributions to global deviations from independence or homogeneity. The methodology proves to be effective in identifying both global and localised departures from the null hypotheses, even in scenarios with subtle mark structures or small sample sizes. Real-world environmental applications to forestry and earthquake data demonstrate the utility of the proposed framework for detecting spatially dependent marked structures in the patterns.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on Gaussian processes, kernel machines, linear models, or linearized neural approximations, leaving a gap between theory and the nonlinear models used in practice. We develop a kernel based framework for analyzing regularized nonlinear parametric models trained on adaptively collected data. Our approach uses kernels over the parameter space to induce reproducing kernel Hilbert space structures over the corresponding model class, yielding confidence bounds for models trained with broad classes of regularized convex losses. We show how these bounds can support convergence guarantees for nonlinear acquisition and surrogate models, including randomized regularized policies that select points by maximizing a trained random model. These results provide a unified route to analyzing nonlinear parametric models in Bayesian optimization and related adaptive optimization settings.

Discrete automated processes in industrial and cyber-physical systems often exhibit a repetitive structure in which successive repetitions follow a common trajectory while differing in duration, amplitude, and fine-scale dynamics. Such \emph{approximately periodic} behavior poses a challenge for Gaussian Processes (GP) modeling: strictly periodic models suppress inter-repetition variability, while non-periodic models fail to capture the strong structural regularities required for generation. In this work, we propose a stochastic generative model for approximately periodic time series. The model is based on a GP whose posterior is modulated by a novel kernel. Our approach decouples intra-repetition structure from inter-repetition variability through a two-stage construction which yields a generative distribution with a identical mean function across repetitions, while allowing smooth variation between repetitions. The modeling choices are supported by an implementation in which realistic synthetic trajectories are generated from toy datasets.

Artificial intelligence (AI) has transformed imaging inverse problems, from medical diagnostics to Earth observation. Yet deep neural networks can produce hallucinations, realistic-looking but incorrect details, undermining their reliability, especially when ground truth data is unavailable. We develop a theoretical framework showing that such hallucinations are not merely artifacts of particular models, but can arise from the ill-posed nature of the inverse problem itself. We derive necessary and sufficient conditions for hallucinations, together with computable bounds on their magnitude that depend only on the forward model. Building on this theory, we introduce algorithms to: (1) estimate the minimum hallucination magnitude achievable by any reconstruction model for a given input; (2) assess the faithfulness of reconstructed details by a given reconstruction model. Experiments across three imaging tasks demonstrate that our approach applies broadly, including to modern generative models, and provides a principled way to quantify and evaluate AI hallucinations.

This paper introduces an algorithm-agnostic approach to feature-based time series clustering via amortized neural inference. By training neural networks to approximate the optimal partitioning rule from simulated data, the proposed framework reduces reliance on conventional clustering methods, such as $K$-means, $K$-medoids, or hierarchical clustering, and their associated objective functions and heuristics. Leveraging statistical features, such as autocorrelations and quantile autocorrelations, the approach learns a data-driven affinity structure from which clustering partitions can be recovered, without requiring explicit prior specification of cluster shapes or structures. In addition, one version of the method can automatically determine the number of clusters, avoiding ad-hoc selection procedures. Comprehensive empirical studies show that the proposed framework achieves competitive or superior clustering accuracy relative to traditional methods, even in challenging scenarios where competing techniques are provided with the true number of clusters. An application to financial time series of stock returns illustrates its practical utility. By reducing the need for algorithm selection and calibration, the proposed framework opens new possibilities for automated, adaptive, and data-driven clustering of temporal data across scientific and industrial domains.

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

Density estimation in high-dimensional settings is an important and challenging statistical problem.Traditional methods based on kernel smoothing are inefficient in high dimensions due to the difficulties in specifying appropriate location-adaptive kernels. In this work, we introduce pre-training, a key idea behind many cutting-edge AI technologies, to the context of non-parametric density estimation. By establishing a pre-trained neural network that can recommend an appropriate location-adaptive kernel for each sample point, efficient density estimation with adaptive kernels is achieved in high dimensions. A wide range of numerical experiments show that this strategy is highly effective for improving density-estimation accuracy, when the target distribution is close to the distribution family for pre-training. When the target distribution is substantially different from the pre-training distribution family, the benefit from the proposed pre-training strategy may be diluted, but can be reactivated by an additional fine-tuning procedure.

We study what remains detectable about one-sided Poisson cluster processes after cluster orientation is erased. We construct matched reversible cluster nulls preserving intensity and the full Bartlett spectrum, showing that second-order structure alone need not identify temporal direction. For stationary Poisson branching clusters, we derive the Fourier--Stieltjes transform of the reduced third cumulant and show that, in the $L^1$ third-cumulant regime, a nonzero imaginary factorial bispectrum certifies orientation. We also give explicit orientation-erased nulls, reversible spectral matches for monotone Hawkes kernels, and finite-window third-order orientation contrasts.

Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

LLM-enabled AI workflows increasingly produce outputs through iterative generate-evaluate-revise loops. Each iteration can improve the candidate, but it also creates a release decision: when to stop and output the current result? This raises a statistical challenge because deployment-time evaluator scores are adaptively generated and repeatedly monitored, yet the likelihood models or exchangeability assumptions typically used for calibration are unavailable. We propose an always-valid release wrapper for existing generator-evaluator pipelines. The wrapper builds a hard-negative reference pool of high-scoring failures, calibrates deployment-time evaluator scores against this pool, and accumulates the resulting evidence with an e-process. This separates two roles: the reference pool turns black-box scores into conservative evidence, while the e-process provides validity under optional stopping. In theory, we show that a conservative reference pool yields finite-sample control of the probability of releasing on infeasible tasks, that is, tasks for which the given workflow is not capable of producing a reliable solution. We also characterize conditions under which the same conservative rule still achieves nontrivial release on feasible tasks. In an MBPP+ coding-agent case study, the wrapper reduces premature incorrect release relative to baseline stopping rules while still releasing on tasks for which the workflow repeatedly accumulates moderate supporting evidence.

Weak-to-strong (W2S) generalization, in which a strong model is fine-tuned on outputs of a weaker, task-specialized model, has been proposed as an approach to aligning superhuman AI systems. Existing theoretical analyses either fix the student's representations or operate in restricted settings. Whether multi-step SGD can succeed in feature learning while preserving diverse pre-trained capabilities remains open. We study W2S in the setting of reward-model learning with two-layer neural networks. The strong model has pre-trained representations organized into low-dimensional subspaces $V_k$, and is fine-tuned under the supervision of a weak model specialized on task $κ$. We prove that the strong model efficiently learns task $κ$, eliciting its pre-trained knowledge while retaining general capabilities. This establishes W2S generalization in the feature-learning regime, in the sense that the strong model acquires the target feature direction through W2S training, rather than having it given a priori. Moreover, W2S preserves pre-trained off-target features, whereas standard supervised fine-tuning causes catastrophic forgetting when off-target feature directions are correlated with the target's. Numerical experiments on synthetic data confirm our theoretical results.

Replicated weighted networks often exhibit many structural zeros alongside heterogeneous non-zero edge strengths. In structural connectomics, this zero-inflation coincides with subjects expressing overlapping, rather than discrete, connectivity patterns. To address these features, we propose a Bayesian adaptive latent mixture model for zero-inflated weighted networks. Our approach represents each subject network as a simplex mixture of shared low-rank latent score matrices, integrated with a hurdle likelihood that separates edge existence from conditional edge strength. A sparsity-coupling parameter enables absent edges to be either independent of, or informative about, the latent connectivity. For computation, we employ transformed Hamiltonian Monte Carlo on unconstrained coordinates, selecting the number of templates via predictive fit, held-out link prediction, and template stability. Theoretically, we establish posterior consistency, local asymptotic normality, a Bernstein--von Mises approximation, and predictive consistency for an identifiable quotient-space estimand under a fixed-template scenario. Simulations demonstrate performance gains over topology-only baselines in settings with mixed memberships or structure-informed sparsity. Applied to Human Connectome Project data, the model recovers stable latent score patterns and heterogeneous subject-level mixtures, with behavioural analyses serving strictly as exploratory annotations rather than confirmatory biomarker claims.

Experimental design has emerged as a powerful approach for improving the sample efficiency of A/B testing, yet existing designs rely critically on correctly specified models. We study robust sequential experimental design under model misspecification and develop a unified framework that covers both contextual bandit and dynamic settings. Theoretically, we prove that our design bounds the worst-case mean squared error of the estimated treatment effect. Empirically, we demonstrate the effectiveness of the proposed approach using synthetic and real-world datasets from a leading technology company.

The rapid advancement of large language models (LLMs) has made machine-generated text increasingly difficult to distinguish from human-written text. While recent studies explore leveraging internal representations of language models to uncover deeper detection signals, these raw features often exhibit substantial overlap between classes, limiting their discriminative power. To address this challenge, we propose Steer-to-Detect (\texttt{S2D}), a two-stage framework for detecting LLM-generated text. In the first stage, \texttt{S2D} learns a steering vector that is injected into the hidden states of a frozen observer LLM, producing representations with improved class separability. In the second stage, detection is performed via a hypothesis testing procedure based on the steered representations. We establish finite-sample, high-probability guarantees for Type I and Type II errors, providing a theoretical characterization of the procedure. Empirically, \texttt{S2D} achieves strong and consistent performance across a range of settings, including out-of-distribution scenarios and adversarial perturbations.

We consider the joint estimation of change point locations and the sparsity pattern of the variance covariance matrix, which is assumed to evolve in a piecewise constant manner. By applying Group Fused LASSO and LASSO penalties to the squared Frobenius norm, we estimate both the covariance structure and the change points. Adaptive weights are incorporated into the penalty terms to enhance change point detection and covariance estimation accuracy. We establish the conditions under which the estimated change points and the sparse estimators within each segment are consistent. To solve the resulting optimization problem efficiently, we develop an alternating direction method of multipliers (ADMM) whose updates reduce to computationally tractable subproblems. The performance of the proposed method is illustrated through synthetic and real data experiments, including comparisons with several competing procedures.

In the classic potential outcome framework, the local average treatment effect (LATE) and its identification via an instrumental variable are stated in a deterministic setting at the individual level: each individual has settled potential outcomes such as ``cured if treated''. Several authors have proposed working instead with \emph{stochastic} potential outcomes -- counterfactual probabilities of the form ``the chance of being cured if treated'' -- but the integration of stochastic potential outcomes with the LATE machinery raises an issue. It is a metaphysical issue: in a stochastic setting, the standard joint-probability definitions of compliers and the LATE assume what I will call the \emph{unique-parallel-universe view}, which asserts that, in any genuinely possible state of the world, every counterfactual condition settles a unique determinate outcome even when the underlying causal disposition is irreducibly chancy. The statistician Dawid (2000) doubts the plausibility of this view; the philosopher Lewis (1973) develops a reductio argument against it. I propose a fully stochastic update to the Rubin causal model that drops the assumption of the unique-parallel-universe view: stochastic potential outcomes are introduced as Bernoulli parameters in their own (small) probability spaces, and are connected to observables via the factorization rule of a causal Bayes net. Within this framework, I define a Degree-of-compliance-weighted Average Treatment Effect (DATE) and prove that, under assumptions analogous to those used for the LATE but rewritten for the fully stochastic setting, the DATE equals the usual IV estimand. The classic LATE identification result emerges as a deterministic special case. Existing IV practice can therefore be reinterpreted: it has been estimating the DATE all along, in a general stochastic setting, without assuming the unique-parallel-universe view.

We use Array-RQMC sampling in a walk on spheres (WOS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from $57$-fold to $2290$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WOS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al.\ (2025) that the mean dimension of Array-RQMC-WOS errors is much higher than an analogous Array-MC-WOS algorithm has.

Marketplace platforms routinely evaluate pricing and allocation policies using logged observational data, yet strong offline performance does not imply that a policy is safe to deploy. In real-time bidding (RTB) marketplaces, reserve-price and floor-policy changes affect not only revenue but also fill, advertiser value, budget pacing, and competition across auctions, creating feedback and interference. The central problem is therefore not to estimate whether a policy improves an offline metric, but to determine whether the available evidence justifies direct launch or only further validation. In this regard, we propose a support-aware decision-support system (DSS) that distinguishes promising from actionable evidence. The framework integrates replay, support-aware off-policy evaluation (OPE), conservative lower-bound ranking, multi-sided guardrails, out-of-time validation, sensitivity analysis, and interference-aware validation design into a claim-preserving pipeline that outputs a launch-readiness classification rather than a single performance estimate. Applying the framework to iPinYou-style RTB logs, we identify a margin-gated floor policy as the leading candidate, with a 47.7% replay yield lift, a 45.8% conservative lower-tail lift, and stable out-of-time performance. However, the framework does not recommend direct launch. A decision-rule ablation shows that simplified pipelines select the same policy but incorrectly recommend deployment, leaving key causal assumptions unresolved. In contrast, the proposed DSS selects the same policy but changes the action to online validation, reflecting missing evidence on propensities, bidder response, and interference. Overall, the contribution is a reproducible DSS protocol that prevents decision overclaim under partial identification and converts offline evaluation into an auditable, action-oriented recommendation.

Single-arm trials are an important study design for evaluating drug efficacy and safety without enrolling patients into a control arm. Although they do not provide the gold-standard evidence of randomized controlled trials, they are increasingly used in clinical development as they offer an efficient, ethical, and practical alternative. A wide variety of approaches can be used to construct control comparators and estimate treatment effects, from fixed comparators informed by clinical knowledge to data-based and model-based patient-level comparators, also known as synthetic controls. Powerful and flexible machine learning models can allow outcome-model-based synthetic controls to overcome key limitations of direct data-based approaches, yield more robust estimates of treatment effects, and provide a principled way to incorporate corrections or encode additional assumptions when external data are not directly comparable. In this work, we argue that outcome-model-based synthetic control arms are an important tool for single-arm trials. We focus on digital twins, personalized predictions of disease progression generated from machine learning models trained on historical datasets, which naturally leverage these flexible approaches. We review doubly robust estimators, present power and sample size formulas, and discuss trade-offs in selecting historical data for training and analysis. We also outline practical considerations for deploying digital twins within the framework of recent FDA draft guidance on the use of artificial intelligence in drug development. Finally, we reanalyze data from trials in amyotrophic lateral sclerosis and Huntington's disease to demonstrate the proposed methods.

Convergence diagnosis for Markov chain Monte Carlo is a matter of fundamental importance in computational statistics: it determines the resources allocated to a particular sampling problem and influences the practitioner's view of the quality of estimates obtained from a Markov chain. Motivated by this, we contribute to the emerging class of coupling-based convergence diagnostic algorithms. Concretely, we study coupling multiple Metropolis-Hastings chains using multi-marginal coupling. We introduce a natural objective for this setting and establish lower and upper bounds by drawing connections to list-level distribution coupling and distributed pairwise-matching problems. This analysis ultimately leads to a shared-randomness Poisson Monte Carlo construction for coupling multiple Markov chains. In this process, we avoid a key dimension-dependent bottleneck in the runtime complexity of classical Poisson Monte Carlo by developing an adaptive rule for updating the point process, yielding significant gains in high-dimensional settings. Experiments on grand couplings of Markov chains show that our methods improve coalescence rates across dimensions, reducing meeting times by up to 50% compared with existing baselines.

Sample size reestimation can be a powerful tool to ensure that a clinical trial meets its prespecified power requirements when uncertainty regarding a design parameter exists at the planning stage. However, long term primary endpoints can be harmful to the efficiency of this trial design. If recruitment is continued while treatment outcomes are awaited, long delay can potentially lead to a large number of pipeline participants being recruited in the trial that do not contribute to the interim analysis. This may lead to a larger number of recruited participants than are actually deemed required, resulting in an overpowered trial with high cost. This paper studies the exact impact of such outcome delay on the efficiency of internal pilot type SSR designs. The distribution of the final sample size post SSR is obtained under various delay lengths for both continuous and binary outcome data, how delay impacts the precision of the final sample size estimate is then discussed. Precisely, the impact of delay on this precision is assessed through RMSE, as well as two more novel metrics, termed the delay impact and cost. The results indicate that with increase in delay length, the delay impact increases, inflating average sample size and power. However, the severity of the effect of delayed outcomes depends highly on the exact trial setting. Trials where the reestimated sample size is smaller than originally planned suffer the most from delayed outcomes, often leading to an overpowered trial. However, the impact of delay is substantially less if the original planned sample size remains smaller than the reestimated sample size.

Calibrated probability outputs of trained classifiers are increasingly used as inputs to downstream regression estimands such as effects, prevalences, or disparities for a latent group observed only on a small labelled subset. A standard practice is to threshold the calibrated score at a confidence cutoff and treat the hard label as the truth. Building on a recent identification result for the underlying moment equation, we develop a calibration-aware diagnostic apparatus for pseudo-labelling pipelines. We derive a closed-form expression for the attenuation bias that confidence thresholding induces in the downstream regression coefficient, and show that the bias can be predicted, before any inference is run, from the residual score variance $V^{*}=\mathbb{E}[\operatorname{Var}(p\mid X)]$ on the unlabelled set after partialling out the downstream controls $X$. We further obtain a sharp sensitivity bound under bounded calibration drift, and identify the boundary $V^{*}=0$, which holds iff $p$ is a deterministic function of $X$; this motivates a structural separation between classifier features $W$ and downstream controls $X\subsetneq W$. Five controlled simulations and a UCI Adult illustration trace the predictions. The contribution is operational: a $(V^{*}, κ)$ decision rule that practitioners can compute from any classifier output to decide whether confidence thresholding is safe.

The block maximum method, which is widely used in extreme value analysis, uses a generalized extreme value distribution to approximate that of the maximum of m observations. The quality of this approximation depends on the value of m and may be poor if m is too small. Surprisingly little attention has been paid to the choice of the block length, although a good choice is crucial to the success of the method. In this paper we assess the effect of taking excessively long blocks in terms of asymptotic relative efficiency, and propose likelihood-based approaches and graphical diagnostics to determine whether a proposed block length is suitable, allowing for potential rounding and left-censoring of observations. We investigate our ideas using simulation and illustrate them using wind speed, river flow and rainfall data.

Large language models (LLMs) are pretrained by minimizing the cross-entropy loss for next-token prediction. In this paper, we study whether this optimization strategy can induce geometric structure in the learned model weights and context embeddings. We approach this problem by analyzing a constrained layer-peeled optimization program, which serves as a mathematically tractable surrogate for LLMs by treating the output projection matrix and last-layer context embeddings as optimization variables. Our analysis of this nonconvex optimization program demonstrates that symmetries in the target next-token distributions are transferred to the global minimizers of the layer-peeled model in a precise group-theoretic sense. Specifically, we prove that when the target tokens exhibit a cyclic-shift symmetry (such as the seven days of the week or the twelve months of the year), the optimal logit matrix is exactly circulant, and the Gram matrices of both the output projections and the context embeddings form circulant geometries as well. Next, for exchangeable target distributions invariant under the symmetric group and, more generally, under two-transitive group actions, we show that the global optimal output projection matrix forms a simplex equiangular tight frame, while the optimal logit matrix and context embeddings inherit the permutation symmetries present in the input data. A key technical step is to reduce the constrained nonconvex factorized problem to an explicit logit-level convex characterization for cyclic symmetry and to a symmetry-based lower bound for permutation symmetry, together with a sharp characterization of the optimal factorization. Finally, we empirically demonstrate that open-source LLMs naturally exhibit symmetries consistent with our theoretical predictions, despite being trained without any explicit regularization promoting such geometric structure.

Given a generalist model, learning a task-relevant specialist representation is fundamental for downstream applications. Identifiability, the asymptotic guarantee of recovering the ground-truth representation, is critical because it sets the ultimate limit of any model, even with infinite data and computation. We study this problem in a completely nonparametric setting, without relying on interventions, parametric forms, or structural constraints. We first prove that the structure between time steps and tasks is identifiable in a fully unsupervised manner, even when sequences lack strict temporal dependence and may exhibit disconnections, and task assignments can follow arbitrarily complex and interleaving structures. We then prove that, within each time step, the task-relevant latent representation can be disentangled from the irrelevant part under a simple sparsity regularization, without any additional information or parametric constraints. Together, these results establish a hierarchical foundation: task structure is identifiable across time steps, and task-relevant latent representations are identifiable within each step. To our knowledge, each result provides a first general nonparametric identifiability guarantee, and together they mark a step toward provably moving from generalist to specialist models.

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$α$ sequential test can be obtained by thresholding an e-process at $1/α$. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.

We study the evaluation of real-valued point predictors under the decision-theoretic framework of mean-consistent loss functions given by the Bregman divergences. We first derive a new version of Murphy's decomposition of the expected loss which does not directly include the response itself but only its predictors. We then relate the miscalibration and the discrimination component of the Murphy's decomposition to Lorenz-curve-based accuracy measures that are widely used in practice. Besides the usual area between the concentration and Lorenz curves, ABC, we introduce a mean-squared version ABC$^2$ that mitigates some of the weaknesses of the original ABC in identifying mean-calibration. More importantly, both ABC and ABC$^2$ are shown to rely on predictor-dependent weights, so they fail to align with the class of mean-consistent scoring functions. In the same spirit, we derive a similar result for the widely used Gini score. These results indicate that ABC, ABC$^2$ and Gini scores may lead to dishonest evaluation of point predictions when used for model selection; this gives support to use mean-consistent loss functions as well as the miscalibration and the discrimination measure from the Murphy's decomposition of the expected loss for model evaluation. Finally, we study forecast dominance when Lorenz curves intersect. We show that Lorenz and Murphy's curves have the same number of crossings and, in the one-crossing case, we establish weaker dominance criteria for subclasses of Bregman divergences through third-degree stochastic dominance.

Conformal prediction provides a principled framework for uncertainty quantification with finite-sample coverage guarantees. While recent work has extended conformal prediction to online and sequential settings, existing methods typically focus on a single coverage level and do not ensure consistency across multiple confidence levels. In many real-world applications, such as weather forecasting, macroeconomic prediction, and risk management, different users operate under heterogeneous risk tolerances and require calibrated uncertainty estimates across a range of coverage levels. In such settings, it is desirable to produce prediction sets corresponding to different coverage levels that are nested and valid simultaneously. In this paper, we propose two novel online conformal prediction methods that output \emph{nested prediction sets} across a range of coverage levels, enabling simultaneous uncertainty quantification across the entire risk spectrum. Beyond interpretability, jointly estimating multiple coverage levels is known to improve statistical efficiency in classical quantile regression by enforcing non-crossing constraints and sharing information across quantiles. Our approaches leverage an online optimization perspective with small regret that translates to quantile estimation error control while enforcing nestedness of prediction sets. Empirical results on synthetic and real-world datasets, including applications in forecasting tasks with heterogeneous risk requirements, demonstrate that our method achieves stable coverage across all levels, strictly nested prediction sets, and improved efficiency compared to existing online conformal baselines.

Reinforcement learning agents for portfolio management are typically trained and deployed as static policies, with no mechanism for using price forecasts at inference time. We propose $\text{FPILOT}$ (**Fin**ancial **P**lugin **I**nference-time **L**earning for **O**ptimal **T**rading), a plugin inference-time optimization framework inspired by Model Predictive Control (MPC). Our key structural insight is that future prices mostly do not depend on one agent's portfolio allocation, so a suitable predictive model can produce a multi-step price trajectory without iterative action-conditioned rollouts as in typical reinforcement learning. At each decision step, we use the forecaster's predicted price trajectory to construct an allocation-based imagined return objective, and optimize the policy at inference-time before executing one step of the trade. Our framework is compatible with any pre-trained agent and adapts the policy to the forecaster's predictions without any retraining. Evaluated across five policy learning algorithms on the TradeMaster DJ30 benchmark, $\text{FPILOT}$ produces consistent improvements in total return and return-based risk-adjusted metrics (Sharpe, Sortino, Calmar), with stochastic policies benefiting more than deterministic ones. Further, using synthetic forecasts at calibrated quality levels, we show that gains consistently improve with forecaster quality, suggesting that our performance will improve based on advances in financial forecasting.

We establish the first population risk bounds for Kolmogorov-Arnold Networks (KANs) trained by mini-batch SGD with gradient clipping, covering non-private SGD as well as differentially private SGD (DP-SGD) with Gaussian perturbations that interpolate between independent and temporally correlated noise. This setting is substantially closer to practice than prior KAN theory along two axes: training is by mini-batch SGD, the standard recipe for modern networks, rather than full-batch gradient descent (GD); and correlated-noise mechanisms have empirically shown a more favorable privacy-utility tradeoff than independent-noise mechanisms. Our results cover the corresponding full-batch GD and independent-noise DP-GD results for KANs by Wang et al. (2026), while yielding sharper fixed-second-layer specializations. The technical core is a new analysis route for correlated-noise DP training in the non-convex regime. Temporal dependence breaks the conditional-centering structure underlying standard one-step SGD arguments, and the projection step obstructs the exact cancellation structure of correlated perturbations. We address these difficulties through an auxiliary unprojected dynamics, a shifted iterate that absorbs the current noise perturbation, and a high-probability bootstrap certifying projection inactivity. Combining this optimization analysis with a stability-based generalization argument yields the stated population risk bounds. To the best of our knowledge, this is the first optimization and population risk analysis of a correlated-noise mechanism for DP training beyond convex learning, in particular for neural networks.

Bayesian inference provides principled uncertainty quantification, but accurate posterior sampling with MCMC can be computationally prohibitive for modern applications. Variational inference (VI) offers a scalable alternative and often yields accurate predictive distributions, but cheap variational families such as mean-field (MF) can produce over-concentrated approximations that miss posterior dependence. We propose variational predictive resampling (VPR), a scalable posterior sampling method that exploits VI's predictive strength within a predictive-resampling framework to better approximate the Bayesian posterior. Given a prior-likelihood pair, VPR repeatedly imputes future observations from the current variational predictive, updates the variational approximation after each imputation, and records the parameter value implied by the completed sample. We establish conditions under which the law of the parameter returned by VPR is well defined and show that its finite-horizon approximation converges to this limit. In a tractable Gaussian location model, we show that VPR with MF variational predictives converges to the exact Bayesian posterior, whereas the optimal MF-VI approximation retains a non-vanishing asymptotic gap. Experiments on linear regression, logistic regression, and hierarchical linear mixed-effects models demonstrate that VPR substantially improves posterior uncertainty quantification and recovers posterior dependence missed by MF-VI, while remaining computationally competitive with, and often more efficient than, MCMC.

We study the sample complexity of empirical plug-in estimation for the powered even-order Gromov-Wasserstein functional between compactly supported probability measures on $\mathbb{R}^{d_x}$ and $\mathbb{R}^{d_y}$. For every fixed pair of integers $r,k\geq 1$, we prove that the two-sample empirical error is bounded at the rate $n^{-2/\max\{\min\{d_x,d_y\},4\}}$, up to a logarithmic factor in the critical case $\min\{d_x,d_y\}=4$. This extends the known quadratic Euclidean upper rate to the full powered even-order family. The proof uses a polynomial decomposition of the even-order GW functional, a generalized duality formula reducing the coupling-dependent term to a compact family of ordinary optimal transport problems, and entropy estimates for semiconcave dual potentials.

Every adaptive learning system must alternate between two operations: consolidating what it already knows and expanding into new evidence. We propose \emph{Consolidation-Expansion Operator Mechanics} (OpMech), a framework that makes this structure precise. The central object is the \emph{order-gap} $\Ogap(θ; e)$, the degree to which a consolidation operator~$Q$ and an expansion operator~$P_e$ fail to commute at a given knowledge state. Because the order-gap is computable from the system's own trajectory, it serves as a real-time control signal: large values indicate that the system is still sensitive to the ordering of consolidation and expansion; once the order-gap falls and stays small, further processing is unlikely to change the outcome. Three results give the signal precise meaning: the order-gap decays along convergent trajectories; a persistently large order-gap implies the system is far from its settled state; and an order-gap-based stopping rule terminates with provable guarantees in both noiseless and bounded-noise settings. The framework applies across five domains: bandits, reinforcement learning, stochastic optimization, continual learning, and recursive language models. We give conditions under which the order-gap reliably tracks convergence in three representative cases. We develop the recursive language model application in detail, showing how OpMech replaces heuristic stopping rules and fixed recursion budgets with principled, evidence-driven alternatives.

\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point. For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

Distributed reinforcement learning trains on data from stale, buggy, or mismatched actors, producing actions with high surprisal (negative log-probability) under the learner's policy. The core difficulty is not surprising data per se, but \emph{negative learning from surprising data}. High-surprisal failures can dominate finite-batch updates through large perpendicular components, while high-surprisal successes reveal opportunities the current policy would otherwise miss. The \textit{Delightful Policy Gradient} (DG) separates these cases by gating each update with delight, the product of advantage and surprisal, suppressing rare failures and preserving rare successes without behavior probabilities. In a tabular analysis, DG suppresses the perpendicular second moment of high-surprisal failures by a policy-overlap factor that vanishes as the learner improves. The advantage sign is essential for surprisal-based filtering: any learner-probability-only gate that suppresses rare failures also suppresses rare successes. On MNIST with simulated staleness, DG without off-policy correction outperforms importance-weighted PG with exact behavior probabilities. On a transformer sequence task with staleness, actor bugs, reward corruption, and rare discovery, DG often achieves nearly order-of-magnitude lower error. When all four frictions act simultaneously, its sample-efficiency advantage is order-of-magnitude and grows with task complexity.

Process capability indices such as $C_{pk}$ are widely used in manufacturing to support supplier qualification, pilot-build release, and production approval. In practice, approval decisions are often based on deterministic threshold rules of the form $\widehat{C}_{pk} \ge C_0$. Because $\widehat{C}_{pk}$ is estimated from finite samples, however, such decisions are inherently stochastic, especially when the true capability lies near the approval threshold. This paper develops a risk-calibrated decision framework for process capability approval that explicitly accounts for estimation uncertainty and asymmetric operational loss. Capability approval is formulated as a binary statistical decision problem, leading to a rule of the form $\widehat{C}_{pk} \ge C_0 + k\,SE(\widehat{C}_{pk})$, where the calibration constant $k$ is determined either by a tolerable failure probability or by a false-accept/false-reject cost ratio. The resulting formulation unifies several commonly used procedures, including deterministic thresholding, lower confidence bound rules, and probability-based approval rules, and naturally extends them to cost-sensitive decision rules derived from asymmetric operational loss. Simulation experiments and an industrial case study show that risk calibration primarily affects near-threshold decisions, improves approval stability, and can substantially reduce expected operational loss when false acceptance is more costly than false rejection.

Neural networks, particularly message-passing neural networks (MPNNs), are increasingly used as heuristics for hard combinatorial optimization problems. Yet many learning-based methods rely on supervision, reinforcement learning, or gradient estimators, causing high computational cost, unstable training, or limited guarantees. Classical approximation algorithms provide worst-case guarantees but are non-differentiable and cannot adapt to structure in natural input distributions. We study this tradeoff through Uniform Facility Location (UniFL), a problem with applications in clustering, summarization, logistics, and supply chains. We propose a fully differentiable MPNN that incorporates approximation-algorithmic principles without solver supervision or discrete relaxations. The model has provable approximation guarantees and empirically improves on standard approximation algorithms, narrowing the gap to integer linear programming.

Missing data imputation, where a model is trained on observed data to estimate unobserved values, is a fundamental problem in machine learning. In this paper, we rigorously formulate imputation model learning as a mean-squared error risk minimisation problem. We show that when the probability of missingness depends on the data, many state-of-the-art methods fail to account for the resulting distribution shift between the observed data used for training and the full data distribution used for evaluation. Consequently, these approaches do not minimise mean-squared error on the full data distribution. Instead, we propose a novel imputation algorithm designed to learn an imputation model from the observed data while explicitly accounting for this distribution shift. Simulation studies show consistent improvements over otherwise identical uncorrected baselines, with average reductions of 3% in RMSE and 7% in Wasserstein distance.

Many engineering and scientific workflows rely on expensive black-box evaluations, requiring sequential decisions that must both improve task performance and reduce uncertainty. Bayesian optimization (BO) and Bayesian experimental design (BED) provide powerful but largely separate treatments of goal-directed optimization and information-seeking experimentation, leaving limited guidance for hybrid settings in which learning and optimization are intrinsically coupled. We propose Pragmatic Curiosity (PraC), a unified framework for hybrid learning and optimization via active inference. PraC evaluates candidate queries by trading information gain about a task-relevant latent symbol against an expected regret-based potential over outcomes. This formulation exposes three operational design choices: which latent quantity should be clarified, how task value is encoded as regret, and how strongly information gain should be exchanged against pragmatic value. We instantiate PraC across three regimes of increasing complexity: decision-oriented plume monitoring with fixed global symbols and known downstream losses, targeted active search with induced local symbols and evolving coverage goals, and composite Bayesian optimization with hierarchical regret learning under unknown preferences. Across these regimes, PraC reduces downstream decision risk, improves coverage of critical outcome regions, and jointly learns predictive and preference structures without relying on task-specific staging rules.

Generative foundation models trained on tokenized electronic health record (EHR) timelines show promise for clinical outcome prediction via Monte Carlo sampling of simulated future trajectories. However, this approach suffers from three coupled limitations: sparse estimate distributions that poorly differentiate patient risk levels, extreme computational cost, and high sampling variance. We propose two new estimators that leverage next-token probability distributions underutilized by standard Monte Carlo: the Sum of Conditional Outcome Probability Estimator (SCOPE) and Risk Estimation from Anticipated Conditional Hazards (REACH). We prove both are unbiased, that REACH guarantees variance reduction over Monte Carlo for any model and outcome, and that REACH is a Rao-Blackwellization of any naive importance sampling scheme that preserves the non-outcome token distribution. Empirically, across $11$ clinically important outcomes in MIMIC-IV and the UChicago health system, SCOPE and REACH match $100$-sample Monte Carlo accuracy with median token reductions of $2.5\times$ to $3.4\times$ and reductions exceeding $80\times$ for the rarest outcomes, with calibration preserved throughout. Because SCOPE reuses a single sampled pool across an arbitrary number of outcomes at no marginal generation cost while REACH provides a per-task variance guarantee, the two estimators are complementary in deployment and together meaningfully reduce the inference budget required for generative EHR foundation models, particularly for rare, high-impact outcomes in healthcare.

This paper introduces an infinite-dimensional Bayesian framework for acoustic seabed tomography, leveraging wave scattering to simultaneously estimate the seabed and its roughness. Tomography is considered an ill-posed problem where multiple seabed configurations can result in similar measurement patterns. We propose a novel approach focusing on the statistical isotropy of the seabed. Utilizing fractional differentiability to identify seabed roughness, the paper presents a robust numerical algorithm to estimate the seabed and quantify uncertainties. Extensive numerical experiments validate the effectiveness of this method, offering a promising avenue for large-scale seabed exploration.

Advances in generative modeling have recently been adapted to tabular data containing discrete and continuous features. However, generating mixed-type features that combine discrete states with an otherwise continuous distribution in a single feature remains challenging. We advance the state-of-the-art in diffusion models for tabular data with a cascaded approach. We first generate a low-resolution version of a tabular data row, that is, the collection of the purely categorical features and a coarse categorical representation of numerical features. Next, this information is leveraged in the high-resolution flow matching model via a novel guided conditional probability path and data-dependent coupling. The low-resolution representation of numerical features explicitly accounts for discrete outcomes, such as missing or inflated values, and therewith enables a more faithful generation of mixed-type features. We formally prove that this cascade tightens the transport cost bound. The results indicate that our model generates significantly more realistic samples and captures distributional details more accurately, for example, the detection score improves by 51.9\%. Code is available at https://github.com/muellermarkus/tabcascade.

Kolmogorov--Arnold Networks (KANs) have recently emerged as a structured alternative to standard MLPs, yet a principled theory for their training dynamics, generalization, and privacy properties remains limited. In this paper, we analyze gradient descent (GD) for training two-layer KANs and derive general bounds that characterize their training dynamics, generalization, and utility under differential privacy (DP). As a concrete instantiation, we specialize our analysis to logistic loss under an NTK-separable assumption, where we show that polylogarithmic network width suffices for GD to achieve an optimization rate of order $1/T$ and a generalization rate of order $1/n$, with $T$ denoting the number of GD iterations and $n$ the sample size. In the private setting, we characterize the noise required for $(ε,δ)$-DP and obtain a utility bound of order $\sqrt{d}/(nε)$ (with $d$ the input dimension), matching the classical lower bound for general convex Lipschitz problems. Our results imply that polylogarithmic width is not only sufficient but also necessary under differential privacy, revealing a qualitative gap between non-private (sufficiency only) and private (necessity also emerges) training regimes. Experiments further illustrate how these theoretical insights can guide practical choices, including network width selection and early stopping.

Recent work has demonstrated surprisingly good performance of pre-trained LLMs on regression tasks (for example, time-series prediction), with the ability to incorporate expert prior knowledge and the information contained in textual metadata. However we observe major error cascades even in short sequences < ~100 points; these models are also computationally intensive and difficult to parallelise. Marginal LLM predictions do not suffer this issue and are trivially parallelised, but can predict over-broad densities. To address this, we propose combining these densities with a lightweight (diffusion-based) neural process. We show that this combination leads to better-calibrated predictions overall, outputs locally consistent trajectories, and leads to text-conditioned function space selection in the meta-learner. As part of this work we propose a gradient-free (and non-Monte Carlo) method for sampling from a product-of-experts of a score model and an 'expert' (here the LLM predictive densities). We believe this general method is of independent interest as it is applicable whenever an expert can be convolved with a Gaussian in closed form.

Special functions have always played a central role in physics and in mathematics, arising as solutions of nonlinear differential equations, as well as in the theory of branching processes, which extensively uses probability generating functions. The theory of iteration of real functions leads to limit theorems for the discrete-time and real-time Markov branching processes. The Poisson reproduction of particles in real time is analysed through the integration of the Kolmogorov equation. These results are further extended by employing graphical representations of Koenigs functions under subcritical and critical branching mechanisms. The limit conditional law in the subcritical case and the invariant measure for the critical case are discussed, as well. The obtained explicit solutions contain the exponential Bell polynomials and the modified exponential-integral function $\rm{Ein} (z)$.

Transport-based methods have emerged as a leading paradigm for building generative models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through a noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce a novel approach to this task based on Stochastic Interpolants: we iteratively update a transport map between corrupted and clean data samples using only access to the corrupted dataset as well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards a self-consistent transport map that effectively inverts the corruption channel, thus enabling a generative model for the clean data. We refer to the resulting method as the self-consistent stochastic interpolant (SCSI). It (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance on inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions. Our source code is publicly available at https://github.com/modichirag/SCSI

We study the base distribution in chart-based generative models on Riemannian manifolds. Standard methods sample in Euclidean tangent space and then map the sample to the manifold with a chart. This is convenient, but it changes the meaning of distance: the same tangent-space scale can correspond to different geodesic radii, i.e. shortest-path distances from a reference point on the manifold, under different charts, curvatures, and dimensions. Within isotropic, scalar-Jacobian azimuthal charts, we show that no base distribution can simultaneously preserve geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it has a specific form, which we call Radial Compensation (RC). RC chooses the tangent-space base so that the model realizes a user-specified one-dimensional law for the geodesic radius, and leaves the chart available as a numerical preconditioner. This gives more stable training and cleaner curvature estimates, because curvature no longer has to compensate for distortions introduced by the chart. We also introduce balanced exponential charts, which improve conditioning without changing the realized manifold density under RC. This decouples the statistical meaning of the model, the law of the geodesic radius, from its numerical conditioning, which is governed by the chart Jacobian: chart choice becomes a numerical preconditioner rather than a hidden modeling decision. Across manifold variational autoencoders and continuous normalizing flows, RC matches the intended radius behavior, improves numerical stability, and makes learned curvature easier to interpret.

Discrete diffusion models have emerged as a powerful class of models and a promising route to fast language generation, but practical implementations typically rely on factored reverse transitions ignoring cross-token dependencies and degrading few-step performance. We propose Latent-Augmented Discrete Diffusion (LADD), which introduces a learnable auxiliary latent channel and performs diffusion over the joint (token, latent) space. The latent variables provide an intermediate representation expressing joint structure while preserving tractable parameterizations. We instantiate LADD with continuous latents (Co-LADD) and discrete latents (Di-LADD), and study two inference schedules: a joint diffusion that denoises data and latents together, and a sequential diffusion that first resolves latents and then samples tokens conditionally. We derive ELBO-style objectives and analyze design choices that balance latent expressivity with diffusion compatibility. In experiments, LADD models yield improvements on unconditional generation metrics as compared to state-of-the-art masked discrete diffusion baselines, and are effective at lower sampling budgets, where unmasking many tokens per step is desirable.

In many business settings, task-specific labeled data are scarce or costly to obtain, limiting supervised learning on a target task. A classical response is transfer learning (TL). Many TL works study how to transfer information from related sources. We study, for linear regression and classification, when to transfer via sample sharing: in a multi-source setting, we greedily decide from which sources and how many samples to incorporate into the target dataset. Our method uses an accept/reject rule based on a data-dependent estimate of the transfer gain, i.e the marginal decrease in target predictive error, computed conditionally on the observed target samples. We analyze our approach and show that how the derived statistical test enforces positive transfer with high probability. Under additional standard conditions, we also study the transfer gain itself and characterize when transfer is beneficial. Experiments on synthetic and real data show consistent gains over classical and recent strong baselines while avoiding negative transfer.

Recent advances in protein structure prediction, such as AlphaFold, have demonstrated the power of deep neural architectures like the Evoformer for capturing complex spatial and evolutionary constraints on protein conformation. However, the depth of the Evoformer, comprising 48 stacked blocks, introduces high computational costs and rigid layerwise discretization. Inspired by Neural Ordinary Differential Equations (Neural ODEs), we propose a continuous-depth formulation of the Evoformer, replacing its 48 discrete blocks with a Neural ODE parameterization that preserves its core attention-based operations. This continuous-time Evoformer achieves constant memory cost (in depth) via the adjoint method, while allowing a principled trade-off between runtime and accuracy through adaptive ODE solvers. Benchmarking on protein structure prediction tasks, we find that the Neural ODE-based Evoformer produces structurally plausible predictions and reliably captures certain secondary structure elements, such as alpha-helices, though it does not fully replicate the accuracy of the original architecture. However, our model achieves this performance using dramatically fewer resources, just 17.5 hours of training on a single GPU, highlighting the promise of continuous-depth models as a lightweight and interpretable alternative for biomolecular modeling. This work opens new directions for efficient and adaptive protein structure prediction frameworks.

Latent categorical variables are frequently found in deep learning architectures. They can model actions in discrete reinforcement-learning environments, represent categories in latent-variable models, or express relations in graph neural networks. Despite their widespread use, their discrete nature poses significant challenges to gradient-descent learning algorithms. While a substantial body of work has offered improved gradient estimation techniques, we take a complementary approach. Specifically, we: 1) revisit the ubiquitous $\textit{softmax}$ function and demonstrate its limitations from an information-geometric perspective; 2) replace the $\textit{softmax}$ with the $\textit{catnat}$ function, a function composed of a sequence of hierarchical binary splits; we prove that this choice offers significant advantages to gradient descent due to the resulting diagonal Fisher Information Matrix. A rich set of experiments - including graph structure learning, variational autoencoders, and reinforcement learning - empirically show that the proposed function improves the learning efficiency and yields models characterized by consistently higher test performance. $\textit{Catnat}$ is simple to implement and seamlessly integrates into existing codebases. Moreover, it remains compatible with standard training stabilization techniques and, as such, offers a better alternative to the $\textit{softmax}$ function.

We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in [26], to address limitations of the standard Gaussian prior. It has been proved in [26] that, as the number of nodes in the hidden layers grows large, according to a sequential and ordered limit, the law of the output converges weakly to a Gaussian mixture. In this paper, we study the neural network through the lens of the posterior distribution with a Gaussian likelihood. If the random covariance matrix of the infinite-width limit is positive definite under the prior, we identify the posterior distribution of the output in the wide-width limit according to a sequential regime. Remarkably, we provide mild sufficient conditions to ensure the aforementioned invertibility of the random covariance matrix under the prior, thereby extending the results in [8]. Among our results, we present sufficient conditions on some model parameters (the activation function and the associated Lévy measures) which ensure that the sequential limits are independent of the order. We illustrate our findings with examples and numerical simulations.

Many datasets include a small set of variables, such as biomarkers or clinical outcomes, whose relationships to the broader system are of primary scientific interest. Estimating the full network of inter-variable relationships in such settings often obscures local structures around these targets, limiting interpretability. To address this fundamental problem, we introduce local graph estimation, a statistical framework for inferring substructures around target variables. We show that traditional graph estimation methods often fail to recover local structure, and present pathwise feature selection (PFS) as an effective alternative. PFS estimates local subgraphs by iteratively applying feature selection and propagating uncertainty along network paths, providing rigorous finite-sample false discovery control even in settings with mixed variable types and nonlinear dependencies. In four distinct applications spanning environmental and public health, multiomics, brain connectomics, and single-nucleus RNA sequencing, PFS recovers interpretable networks consistent with domain knowledge, highlighting its ability to uncover established mechanisms and generate novel hypotheses.

This paper introduces the framework of multi-armed sampling, which serves as the sampling counterpart to the optimization problem of multi-armed bandits. Our primary motivation is to rigorously examine the exploration-exploitation trade-off in the context of sampling. We systematically define plausible notions of regret for this framework and establish corresponding lower bounds. We then propose a simple algorithm that achieves near-optimal regret bounds. Our theoretical results suggest that, in contrast to optimization, sampling barely requires any exploration. To further connect our findings with those of multi-armed bandits, we define a continuous family of problems and associated regret measures that smoothly interpolate and unify multi-armed sampling and multi-armed bandit problems using a temperature parameter. We believe that the multi-armed sampling framework and our findings in this setting can play a foundational role in the study of sampling, including recent neural samplers, much like the role of multi-armed bandits in reinforcement learning. In particular, our work sheds light on the role of exploration (or lack thereof) and the convergence properties of algorithms for entropy-regularized reinforcement learning, fine-tuning of pretrained models and reinforcement learning with human feedback (RLHF).

Monitoring aboveground biomass (AGB) and its density (AGBD) at high resolution is essential for carbon accounting and ecosystem management. While NASA's spaceborne Global Ecosystem Dynamics Investigation (GEDI) LiDAR mission provides globally distributed reference measurements for AGBD estimation, the majority of commercial remote sensing products based on GEDI remain without rigorous or independent validation. Here, we present an independent regional validation of an AGBD dataset offered by terraPulse, Inc., based on independent reference data from the US Forest Service Forest Inventory and Analysis (FIA) program. Aggregated to 64,000-hectare hexagons and US counties across the US states of Utah, Nevada, and Washington, we found very strong agreement between terraPulse and FIA estimates. At the hexagon scale, we report R2 = 0.88, RMSE = 26.68 Mg/ha, and a correlation coefficient (r) of 0.94. At the county scale, agreement improves to R2 = 0.90, RMSE =32.62 Mg/ha, slope = 1.07, and r = 0.95. Spatial and statistical analyses indicated that terraPulse AGBD values tended to exceed FIA estimates in non-forest areas, likely due to FIA's limited sampling of non-forest vegetation. The terraPulse AGBD estimates also exhibited lower values in high-biomass forests, likely due to saturation effects in its optical remote-sensing covariates. This study advances operational carbon monitoring by delivering a scalable framework for comprehensive AGBD validation using independent FIA data, as well as a benchmark validation of a new commercial dataset for global biomass monitoring.

Compression and generalization are fundamentally related through Solomonoff induction and the minimum description length principle (MDL), which predict that simpler models generalize better when data arises from low-complexity distributions. In this article, we combine insights from algorithmic information theory and techniques from neural network pruning to improve model generalization by identifying the most effective data compression method. Since exact MDL optimization is intractable, we cast it as $\ell_0$ regularized learning and explain why parameter sparsity provides an effective computable approximation of model description length. To identify the best practical approach, we systematically compare and refine complementary sparse optimization methods. In particular, we improve probabilistic pruning through a procedure that does not require Monte Carlo sampling and refine smooth $\ell_0$ approximations with a binary search routine that reduces hyperparameter complexity. Across convolutional networks and transformers evaluated on image and text datasets, our refined methods improve upon their predecessors, achieve substantial model compression with minimal accuracy loss, and yield short data description lengths. Finally, we use these methods in a controlled teacher-student setting to empirically verify the prediction of Solomonoff induction that compressed models learn more sample-efficiently and generalize better.

Bootstrap methods have long been the cornerstone of ensemble learning in machine learning. This paper presents a theoretical analysis of bootstrap techniques applied to the Least Square Support Vector Machine (LSSVM) ensemble in the context of large and growing sample sizes and feature dimensionalities. Using tools from Random Matrix Theory, we investigate the performance of this classifier that aggregates decision functions from multiple weak classifiers, each trained on different subsets of the data. We provide insights into the use of bootstrap methods in high-dimensional settings, enhancing our understanding of their impact. Based on these findings, we propose strategies to select the number of subsets and the regularization parameter that maximize the performance of the LSSVM. Empirical experiments on synthetic and real-world datasets validate our theoretical results.

We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the \emph{equality} of two non-atomic (Borel) probability measures on a locally compact uncountable Polish space is \emph{equivalent} to testing for the \emph{singularity} between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an information-theoretic perspective than non-parametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This ``separation of measure phenomenon'' appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods.

In this work, we propose a new particle-based variational inference (ParVI) method for accelerating the Energetic Variational Inference with Implicit scheme (EVI-Im) introduced in Ref. \cite{wang2021particle}. Inspired by energy quadratization (EQ) and operator splitting techniques for gradient flows, the proposed method efficiently drives particles towards the target distribution, while retaining a meaningful stability mechanism. Unlike EVI-Im, which employs the implicit Euler method to solve variational-preserving particle dynamics obtained from a "discretization-then-variation" approach for minimizing the Kullback--Leibler divergence, the proposed algorithm avoids repeated evaluation of inter-particle interaction terms within each time step, significantly reducing computational cost. The framework is also extensible to other gradient-based sampling techniques. Through several numerical experiments, we demonstrate that the proposed method achieves competitive performance compared with existing ParVI approaches, while offering advantages in efficiency and robustness in certain regimes.

We prove that transformers can exactly interpolate datasets of finite input sequences in $\mathbb{R}^d$, $d\geq 2$, with corresponding output sequences of smaller or equal length. Specifically, given $N$ sequences of arbitrary but finite lengths in $\mathbb{R}^d$ and output sequences of lengths $m^1, \dots, m^N \in \mathbb{N}$, we construct a transformer with $\mathcal{O}(\sum_{j=1}^N m^j)$ blocks and $\smash{\mathcal{O}(d \sum_{j=1}^N m^j)}$ parameters that exactly interpolates the dataset. Our construction provides complexity estimates that are independent of the input sequence length, by alternating feed-forward and self-attention layers and by capitalizing on the clustering effect inherent to the latter. Our novel constructive method also uses low-rank parameter matrices in the self-attention mechanism, a common feature of practical transformer implementations. These results are first established in the hardmax self-attention setting, where the geometric structure permits an explicit and quantitative analysis, and are then extended to the softmax setting. Finally, we demonstrate the applicability of our exact interpolation construction to learning problems, in particular by providing convergence guarantees to a global minimizer under regularized training strategies. Our analysis contributes to the theoretical understanding of transformer models, offering an explanation for their excellent performance in exact sequence-to-sequence interpolation tasks.

Hypergraph data, which capture multi-way interactions among entities, are increasingly prevalent in the big data era. Generating new hyperlinks from an observed, usually high-dimensional hypergraph is an important yet challenging task with diverse applications in areas such as electronic health record analysis and biological research. This task is fraught with several challenges. The discrete nature of hyperlinks renders many existing generative models inapplicable. Additionally, powerful machine learning-based generative models often operate as black boxes, providing limited interpretability. Key structural characteristics of hypergraphs, including node degree heterogeneity and hyperlink sparsity, further complicate the modeling process and must be carefully addressed. To tackle these challenges, we propose Denoising Diffused Embeddings (DDE), a general and efficient generative modeling architecture for hypergraphs. DDE exploits low-rank structure in high-dimensional hypergraphs via a conditional hyperlink likelihood model that links discrete hyperlinks to a continuous latent embedding space and leverages a score-based diffusion model to reconstruct that space. Theoretically, we show that when true latent embeddings are accessible, DDE exactly reduces the task of generating new high-dimensional hyperlinks to generating new low-dimensional embeddings. Moreover, we analyze the implications of using estimated embeddings in DDE, revealing how hypergraph characteristics such as dimensionality, node degree heterogeneity, and hyperlink sparsity impact its generative performance. Simulation studies demonstrate the superiority of DDE over existing methods, in terms of both computational efficiency and generative performance. Furthermore, an application to a symptom co-occurrence hypergraph derived from electronic medical records uncovers interesting findings and highlights the advantages of DDE.

A key question in brain sciences is how to identify time-evolving functional connectivity, such as that obtained from recordings of neuronal activity over time. We wish to explain the observed phenomena in terms of latent states which, in the case of neuronal activity, might correspond to subnetworks of neurons within a brain or organoid. Many existing approaches assume that only one latent state can be active at a time, in contrast to our domain knowledge. We propose a switching dynamical system based on the factorial hidden Markov model. Unlike existing approaches, our model acknowledges that neuronal activity can be caused by multiple subnetworks, which may be activated either jointly or independently. A change in one part of the network does not mean that the entire connectivity pattern will change. We pair our model with scalable variational inference algorithm, using a concrete relaxation of the underlying factorial hidden Markov model, to effectively infer the latent states and model parameters. We show that our algorithm can recover ground-truth structure and yield insights about the maturation of neuronal activity in microelectrode array recordings from in vitro neuronal cultures.

Algorithmic fairness has become a central concern in modern machine learning and AI applications. However, two pressing challenges remain: (1) The fairness guarantees of existing methods often rely on specific data distributional assumptions and large sample sizes, which can lead to fairness violations in practice. (2) Due to legal and societal considerations, using sensitive group attributes during decision-making (referred to as the group-blind setting) may not always be feasible. In this work, we quantify the impact of enforcing algorithmic fairness and group-blindness/awareness in binary classification under group fairness constraints. Specifically, we propose a unified framework for fair classification that provides distribution-free and finite-sample fairness guarantees with controlled excess risk. This framework is applicable to various group fairness notions in both group-aware and group-blind scenarios. Our approach is based on a post-processing procedure that can be applied to arbitrary black-box models, making it directly compatible with modern machine learning pipelines. Furthermore, we establish a minimax lower bound showing the minimax rate-optimality of our proposed algorithm up to logarithmic factors. Through extensive synthetic and real data studies, we further demonstrate the competitive or superior performance of our algorithm compared to existing methods, and provide empirical support for our theoretical findings.

In this paper, we present a new ensemble-based filter method by reconstructing the analysis step of the particle filter through a transport map, which directly transports prior particles to posterior particles. The transport map is constructed through an optimization problem described by the Maximum Mean Discrepancy loss function, which matches the expectation information of the approximated posterior and reference posterior. The proposed method inherits the accurate estimation of the posterior distribution from particle filtering while gives an extension to high dimensional assimilation problems. To improve the robustness of Maximum Mean Discrepancy, a variance penalty term is used to guide the optimization. It prioritizes minimizing the discrepancy between the expectations of highly informative statistics for the reference posteriors. The penalty term significantly enhances the robustness of the proposed method and leads to a better approximation of the posterior. A few numerical examples are presented to illustrate the advantage of the proposed method over ensemble Kalman filter.

Transformers are extremely successful machine learning models whose mathematical properties remain poorly understood. Here, we rigorously characterize the behavior of transformers with hardmax self-attention and normalization sublayers as the number of layers tends to infinity. By viewing such transformers as discrete-time dynamical systems describing the evolution of points in a Euclidean space, and thanks to a geometric interpretation of the self-attention mechanism based on hyperplane separation, we show that the transformer inputs asymptotically converge to a clustered equilibrium determined by special points called \textit{leaders}. We then leverage this theoretical understanding to solve sentiment analysis problems from language processing using a fully interpretable transformer model, which effectively captures `context' by clustering meaningless words around leader words carrying the most meaning. Finally, we outline remaining challenges to bridge the gap between the mathematical analysis of transformers and their real-life implementation.

This paper develops a generative model by minimizing the second-order Wasserstein loss (the $W_2$ loss) through a distribution-dependent ordinary differential equation (ODE), whose dynamics involves the Kantorovich potential associated with the true data distribution and a current estimate of it. A main result shows that the time-marginal laws of the ODE form a gradient flow for the $W_2$ loss, which converges exponentially to the true data distribution. An Euler scheme for the ODE is proposed and it is shown to recover the gradient flow for the $W_2$ loss in the limit. An algorithm is designed by following the scheme and applying persistent training, which naturally fits our gradient-flow approach. In both low- and high-dimensional experiments, our algorithm outperforms Wasserstein generative adversarial networks by increasing the level of persistent training appropriately.

Parametric assumptions such as exponential distribution are commonly used in clinical trial design and analysis. However, violation of distribution assumptions can introduce biases in sample size and power calculations. Piecewise exponential (PWE) hazard model partitions the hazard function into segments each with constant hazards and is easy for interpretation and computation. Due to its piecewise property, PWE can fit a wide range of survival curves and accurately predict the future number of events and analysis time in event-driven clinical trials, thus enabling more flexible and reliable study designs. Compared with other existing approaches, the PWE model provides a superior balance of flexibility and robustness in model fitting and prediction. The proposed PWEXP package is designed for estimating and predicting PWE hazard models for right-censored data. By utilizing well-established criteria such as AIC, BIC, and cross-validation log-likelihood, the PWEXP package chooses the optimal number of change-points and determines the optimal position of change-points. With its particular goodness-of-fit, the PWEXP provides accurate and robust hazard estimation, which can be used for reliable power calculation at study design and timeline prediction at study conduct. The package also offers visualization functions to facilitate the interpretation of survival curve fitting results.

The COVID-19 pandemic has exerted a profound impact on the global economy and continues to exact a significant toll on human lives. The COVID-19 case growth rate stands as a key epidemiological parameter to estimate and monitor for effective detection and containment of the resurgence of outbreaks. A fundamental challenge in growth rate estimation and hence outbreak detection is balancing the accuracy-speed tradeoff, where accuracy typically degrades with shorter fitting windows. In this paper, we provide a transfer learning framework, which we call Transfer Learning Random Forest (TLRF), for an effective implementation of the random forests algorithm that balances this accuracy-speed tradeoff. Specifically, we develop an identification strategy that converts the growth rate estimation problem into a regression task, which enables effective transfer learning across space and time through random forests' adaptive weighting mechanism. As such, through adaptively choosing fitting window sizes based on relevant day-level and county-level features affecting the disease spread, TLRF can accurately estimate case growth rates for counties with small sample sizes. Out-of-sample prediction analysis shows that TLRF outperforms established growth rate estimation methods. Furthermore, we conducted a case study based on outbreak case data from the state of Colorado and showed that TLRF could improve timely detections of outbreaks up to 224% when compared to the decisions made by Colorado's Department of Health and Environment (CDPHE). To demonstrate practical implementation, we developed a publicly available outbreak detection tool that operated from September 2020 through March 2023, receiving substantial attention from policymakers across all 50 states.

Fully conditional specification (FCS) is a convenient and flexible multiple imputation approach. It specifies a sequence of simple regression models instead of a potential complex joint density for missing variables. However, FCS may not converge to a stationary distribution. Many authors have studied the convergence properties of FCS when priors of conditional models are non-informative. We extend to the case of informative priors. This paper evaluates the convergence properties of the normal linear model with normal-inverse gamma prior. The theoretical and simulation results prove the convergence of FCS and show the equivalence of prior specification under the joint model and a set of conditional models when the analysis model is a linear regression with normal inverse-gamma priors.

Missing data are often dealt with multiple imputation. A crucial part of the multiple imputation process is selecting sensible models to generate plausible values for incomplete data. A method based on posterior predictive checking is proposed to diagnose imputation models based on posterior predictive checking. To assess the congeniality of imputation models, the proposed diagnostic method compares the observed data with their replicates generated under corresponding posterior predictive distributions. If the imputation model is congenial with the substantive model, the observed data are expected to be located in the centre of corresponding predictive posterior distributions. Simulation and application are designed to investigate the proposed diagnostic method for parametric and semi-parametric imputation approaches, continuous and discrete incomplete variables, univariate and multivariate missingness patterns. The results show the validity of the proposed diagnostic method.

We apply the procedure of Lee et al. to the problem of performing inference on the signal-noise ratio of the asset which displays maximum sample Sharpe ratio over a set of possibly correlated assets. We find a multivariate analogue of the commonly used approximate standard error of the Sharpe ratio to use in this conditional estimation procedure. We also consider several alternative procedures, including the simple Bonferroni correction for multiple hypothesis testing, which we fix for the case of positive common correlation among assets, the chi-bar square test against one-sided alternatives, Follman's test, and Hansen's asymptotic adjustments. Testing indicates the conditional inference procedure achieves nominal type I rate, and does not appear to suffer from non-normality of returns. The conditional estimation test has low power under the alternative where there is little spread in the signal-noise ratios of the assets, and high power under the alternative where a single asset has high signal-noise ratio. Unlike the alternative procedures, it appears to enjoy rejection probabilities monotonic in the signal-noise ratio of the selected asset, and actually maintains near-nominal rejection rates under the conditional null.

Modeling dependencies between random variables independently from their marginals is fundamental in applications ranging from finance to (structural) biology. In this work, we undertake this problem using circula to model data living on the $d$-dimensional flat torus $\mathbb{T}^d$, making two contributions. First, using a low rank covariance structure to define circulae based on a latent variable model, we design the first closed-form normalized distribution on the flat torus $\mathbb{T}^d$--with covariance structure. Second, building on this framework, we propose the first models for joint distributions of torsion angles (backbone and side-chains) for neighboring amino-acids in proteins. In practice, we fit mixtures on flat torii from $\mathbb{T}^{2}$ to $\mathbb{T}^{14}$, and show they are SOTA in terms of likelihood and sparsity. We anticipate that these models will prove fundamental to move from discrete structural studies like in AlphaFold2, to thermodynamics and kinetics, which are the ultimate goals in theoretical biophysics.

Zador's celebrated theorem is a cornerstone of optimal quantisation, establishing both the weak limit of the empirical distribution of an $n$-point optimal quantiser in $R^d$ and the decay rate of the associated $L_s$-mean quantisation error. However, for large dimensions $d$, observing this asymptotic behaviour demands an astronomically large sample size $n$, which grows super-exponentially with $d$. Through a detailed analysis of the quantisation problem for spherically symmetric distributions, we demonstrate that for moderate $n$ random quantisers uniformly distributed on a sphere of suitable radius $r$ achieve exceptional performance. The expected distortion, expressed as a triple integral, can be computed with arbitrary precision, and the optimal radius $r$ can be efficiently determined numerically. Leveraging results from extreme-value theory, we derive approximations for $r$, particularly in scenarios where $n$ scales with $d$. Depending on the growth rate of $n$, $r$ may either converge to zero or approach a limiting value that is independent of $s$.

Conventional algorithmic trading systems are grounded in deterministic heuristics or offline-trained statistical models that cannot adapt to the semantic complexity of rapidly shifting market regimes. This paper introduces AGENTICAITA, an agentic AI framework that replaces the traditional signal then execute paradigm with a fully autonomous deliberative loop in which multiple specialized Large Language Model agents reason, negotiate, and act in concert - without any offline training or human intervention. The framework proposes four architectural contributions: (i) an Adaptive Z-Score Trigger Engine that acts as a cognitive resource allocator, gating LLM inference exclusively on statistically anomalous market conditions; (ii) a Sequential Deliberative Pipeline - the core agentic contribution - in which an Analyst agent, a Risk Manager agent, and an Executor agent form a structured reasoning chain governed by typed JSON contracts and a deterministic hard-gate safety layer; (iii) an Inference Gating Protocol, a mutex-based cognitive resource scheduler that serializes concurrent agent activations and ensures fully reproducible audit trails; and (iv) a Correlation-Break Diversification composite score that operationalizes portfolio-level idiosyncratic signal prioritization within individual agent reasoning. Validated over a five-day autonomous dry-run session under live market conditions, the framework demonstrates operational correctness of the deliberative pipeline, achieving 157 zero-intervention invocations across 76 assets with an 11.5% agentic friction rate that confirms non-trivial inter-agent negotiation. This preliminary proof-of-concept establishes the feasibility of training-free, deterministic safety-constrained multi-agent orchestration in financial decision loops, with statistically robust performance evaluation and execution cost modeling deferred to extended live deployment.

Scientific innovation increasingly depends on collaboration, yet the organizational structure that fosters breakthrough ideas remains poorly understood. Existing metrics - such as team size or compositional diversity - capture readily observable characteristics but not the deeper architecture of collaboration. We introduce Structural Diversity (SD): the extent to which a team bridges multiple distinct knowledge communities within its prior collaboration network. Using a century-scale dataset of 260 million scientific publications (1900-2025) and combining causal inference with a quasi-natural experiment based on a U.S. National Science Foundation policy change in 2012, we show that SD is a powerful and robust predictor of disruptive innovation, outperforming traditional team novelty indicators such as team freshness and edge density. Moreover, SD positively interacts with team size and is able to mitigate the well-known "curse of scale" by transforming scale from a liability into a resource for creative synthesis. We find that one mechanism underlying this effect is Disciplinary Integration (DI): teams with higher SD can more effectively combine heterogeneous knowledge into novel configurations. Our findings position SD as both a new theoretical construct and an actionable design principle for organizing scientific collaboration. By linking the architecture of team assembly to the dynamics of creative discovery, our work offers a structural explanation for how collective intelligence can be systematically engineered to foster disruptive innovation.

The average treatment effect (ATE), the mean difference in potential outcomes under treatment and control, is a canonical causal effect. Overlap, which says that all subjects have non-zero probability of either treatment status, is necessary to identify and estimate the ATE. When overlap fails, the standard solution is to change the estimand, and target a trimmed effect in a subpopulation satisfying overlap. When the outcome is bounded, we demonstrate that this compromise is unnecessary. We derive non-overlap bounds: partial identification bounds on the ATE that do not require overlap. The bounds have width proportional to the size of the non-overlap subpopulation, making them informative in common scenarios when overlap violations are limited. Since the bounds are non-smooth functionals, we derive smooth approximations amenable to semiparametric efficiency theory and propose a Targeted Minimum Loss-Based estimator that is $\sqrt{n}$-consistent and asymptotically normal under nonparametric conditions. A multiplier bootstrap procedure yields uniformly valid confidence sets across all non-overlap subpopulation sizes and smoothing parameters, allowing researchers to report the tightest valid interval. Formally, we compare non-overlap confidence intervals to confidence intervals based on point estimation across multiple overlap regimes. We illustrate the method via simulation studies and real-world data applications.