This article is an exposition on some recent theoretical advances in learning latent structured models, with a primary focus on the fundamental roles that optimal transport distances play in the statistical theory. We aim at what may be the most critical and novel ingredient in this theory: the motivation, formulation, derivation and ramification of inverse bounds, a rich collection of structural inequalities for latent structured models which connect the space of distributions of unobserved structures of interest to the space of distributions for observed data. This theory is illustrated on classical mixture models, as well as the more modern hierarchical models that have been developed in Bayesian statistics, machine learning and related fields.

We introduce a smooth variant of the SCAD thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on R, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein's unbiased risk estimate is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure.

Gaussian Random Fields (GRFs) with Matérn covariance functions have emerged as a powerful framework for modeling spatial processes due to their flexibility in capturing different features of the spatial field. However, the smoothness parameter is challenging to estimate using maximum likelihood estimation (MLE), which involves evaluating the likelihood based on the full covariance matrix of the GRF, due to numerical instability. Moreover, MLE remains computationally prohibitive for large spatial datasets. To address this challenge, we propose the Fisher-BackTracking (Fisher-BT) method, which integrates the Fisher scoring algorithm with a backtracking line search strategy and adopts a series approximation for the modified Bessel function. This method enables an efficient MLE estimation for spatial datasets using the ExaGeoStat high-performance computing framework. Our proposed method not only reduces the number of iterations and accelerates convergence compared to derivative-free optimization methods but also improves the numerical stability of the smoothness parameter estimation. Through simulations and real-data analysis using a soil moisture dataset covering the Mississippi River Basin, we show that the proposed Fisher-BT method achieves accuracy comparable to existing approaches while significantly outperforming derivative-free algorithms such as BOBYQA and Nelder-Mead in terms of computational efficiency and numerical stability.

Robustness under perturbation and contamination is a prominent issue in statistical learning. We address the robust nonlinear regression based on the so-called interval conditional value-at-risk (In-CVaR), which is introduced to enhance robustness by trimming extreme losses. While recent literature shows that the In-CVaR based statistical learning exhibits superior robustness performance than classical robust regression models, its theoretical robustness analysis for nonlinear regression remains largely unexplored. We rigorously quantify robustness under contamination, with a unified study of distributional breakdown point for a broad class of regression models, including linear, piecewise affine and neural network models with $\ell_1$, $\ell_2$ and Huber losses. Moreover, we analyze the qualitative robustness of the In-CVaR based estimator under perturbation. We show that under several minor assumptions, the In-CVaR based estimator is qualitatively robust in terms of the Prokhorov metric if and only if the largest portion of losses is trimmed. Overall, this study analyzes robustness properties of In-CVaR based nonlinear regression models under both perturbation and contamination, which illustrates the advantages of In-CVaR risk measure over conditional value-at-risk and expectation for robust regression in both theory and numerical experiments.

We derive the unique e-values with optimal (relative) growth rate in the worst case for testing the mean of a bounded random variable, hereby contributing with the first application beyond the assumption of mutually absolutely continuous hypotheses of the (RE)GROW quality criteria for e-values originally proposed by Grünwald et al. (2024). For both criteria, we characterise explicitly the alternatives for which it is most difficult to test against, which also admit a meaningful interpretation. We give two important examples of interest where REGROW provides a powerful quality criterion to choose optimal e-variables whereas GROW leads to trivial solutions.

We study monotone paths in Erdős-Rényi random graphs on numbered vertices. Benjamini & Tzalik established a phase transition at $p = \frac{\log n}{n}$ for this model. We refine the critical value to $p = \frac{\log n - \log \log n }{n}$ and identify the critical window of order $Θ(1/n)$.

An important task in the statistical analysis of inhomogeneous point processes is to investigate the influence of a set of covariates on the point-generating mechanism. In this article, we consider the nonparametric Bayesian approach to this problem, assuming that $n$ independent and identically distributed realizations of the point pattern and the covariate random field are available. In many applications, different covariates are often vastly diverse in physical nature, resulting in anisotropic intensity functions whose variations along distinct directions occur at different smoothness levels. To model this scenario, we employ hierarchical prior distributions based on multi-bandwidth Gaussian processes. We prove that the resulting posterior distributions concentrate around the ground truth at optimal rate as $n\to\infty$, and achieve automatic adaptation to the anisotropic smoothness. Posterior inference is concretely implemented via a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that incorporates a dimension-robust sampling scheme to handle the functional component of the proposed nonparametric Bayesian model. Our theoretical results are supported by extensive numerical simulation studies. Further, we present an application to the analysis of a Canadian wildfire dataset.

We introduce manifolds with kinks, a class of manifolds with possibly singular boundary that notably contains manifolds with smooth boundary and corners. We derive the asymptotic behavior of the Graph Laplace operator with Gaussian kernel and its deterministic limit on these spaces as bandwidth goes to zero. We show that this asymptotic behavior is determined by the inward sector of the tangent space and, as special cases, we derive its behavior near interior and singular points. Lastly, we show the validity of our theoretical results using numerical simulation.

We present Free energy Estimators with Adaptive Transport (FEAT), a novel framework for free energy estimation -- a critical challenge across scientific domains. FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences. Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes a principled foundation for neural free energy calculations. Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates improvements over existing learning-based methods. Our PyTorch implementation is available at https://github.com/jiajunhe98/FEAT.

Existing distribution compression methods, like Kernel Herding (KH), were originally developed for unlabelled data. However, no existing approach directly compresses the conditional distribution of \textit{labelled} data. To address this gap, we first introduce the Average Maximum Conditional Mean Discrepancy (AMCMD), a metric for comparing conditional distributions, and derive a closed form estimator. Next, we make a key observation: in the context of distribution compression, the cost of constructing a compressed set targeting the AMCMD can be reduced from cubic to linear. Leveraging this, we extend KH to propose Average Conditional Kernel Herding (ACKH), a linear-time greedy algorithm for constructing compressed sets that target the AMCMD. To better understand the advantages of directly compressing the conditional distribution rather than doing so via the joint distribution, we introduce Joint Kernel Herding (JKH), an adaptation of KH designed to compress the joint distribution of labelled data. While herding methods provide a simple and interpretable selection process, they rely on a greedy heuristic. To explore alternative optimisation strategies, we also propose Joint Kernel Inducing Points (JKIP) and Average Conditional Kernel Inducing Points (ACKIP), which jointly optimise the compressed set while maintaining linear complexity. Experiments show that directly preserving conditional distributions with ACKIP outperforms both joint distribution compression and the greedy selection used in ACKH. Moreover, we see that JKIP consistently outperforms JKH.

This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an $\ell_1$-penalty that involves some learnable dictionary; and (ii) an analysis form with an $\ell_\infty$-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted $\ell_1$ penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.

We present a general non-parametric statistical inference theory for integrals of quantiles without assuming any specific sampling design or dependence structure. Technical considerations are accompanied by examples and discussions, including those pertaining to the bias of empirical estimators. To illustrate how the general results can be adapted to specific situations, we derive - at a stroke and under minimal conditions - consistency and asymptotic normality of the empirical tail-value-at-risk, Lorenz and Gini curves at any probability level in the case of the simple random sampling, thus facilitating a comparison of our results with what is already known in the literature. Results, notes and references concerning dependent (i.e., time series) data are also offered. As a by-product, our general results provide new and unified proofs of large-sample properties of a number of classical statistical estimators, such as trimmed means, and give additional insights into the origins of, and the reasons for, various necessary and sufficient conditions.

The double descent phenomenon challenges traditional statistical learning theory by revealing scenarios where larger models do not necessarily lead to reduced performance on unseen data. While this counterintuitive behavior has been observed in a variety of classical machine learning models, particularly modern neural network architectures, it remains elusive within the context of quantum machine learning. In this work, we analytically demonstrate that linear regression models in quantum feature spaces can exhibit double descent behavior by drawing on insights from classical linear regression and random matrix theory. Additionally, our numerical experiments on quantum kernel methods across different real-world datasets and system sizes further confirm the existence of a test error peak, a characteristic feature of double descent. Our findings provide evidence that quantum models can operate in the modern, overparameterized regime without experiencing overfitting, potentially opening pathways to improved learning performance beyond traditional statistical learning theory.

This paper introduces a framework for approximate message passing (AMP) in dynamic settings where the data at each iteration is passed through a linear operator. This framework is motivated in part by applications in large-scale, distributed computing where only a subset of the data is available at each iteration. An autoregressive memory term is used to mitigate information loss across iterations and a specialized algorithm, called projection AMP, is designed for the case where each linear operator is an orthogonal projection. Precise theoretical guarantees are provided for a class of Gaussian matrices and non-separable denoising functions. Specifically, it is shown that the iterates can be well-approximated in the high-dimensional limit by a Gaussian process whose second-order statistics are defined recursively via state evolution. These results are applied to the problem of estimating a rank-one spike corrupted by additive Gaussian noise using partial row updates, and the theory is validated by numerical simulations.

Motivated by the empirical observation of power-law distributions in the credits (e.g., ``likes'') of viral posts in social media, we introduce a high-dimensional tail index regression model and propose methods for estimation and inference of its parameters. First, we propose a regularized estimator, establish its consistency, and derive its convergence rate. Second, we debias the regularized estimator to facilitate inference and prove its asymptotic normality. Simulation studies corroborate our theoretical findings. We apply these methods to the text analysis of viral posts on X (formerly Twitter).

This paper studies extremal quantiles under two-way clustered dependence. We show that the limiting distribution of unconditional intermediate-order tail quantiles is Gaussian. This result is notable because two-way clustering typically leads to non-Gaussian limiting behavior. Remarkably, extremal quantiles remain asymptotically Gaussian even in degenerate cases. Building on this insight, we extend our analysis to extremal quantile regression at intermediate orders. Simulation results corroborate our theoretical findings. Finally, we provide an empirical application to growth-at-risk, showing that earlier empirical conclusions remain robust even after accounting for two-way clustered dependence in panel data and the focus on extreme quantiles.

We analyze the frequency spectrum of Quantum Neural Networks (QNNs) using Minkowski sums, which yields a compact algebraic description and permits explicit computation. Using this description, we prove several maximality results for broad classes of QNN architectures. Under some mild technical conditions we establish a bijection between classes of models with the same area $A:=R\cdot L$ that preserves the frequency spectrum, where $R$ denotes the number of qubits and $L$ the number of layers, which we consequently call spectral invariance under area-preserving transformations. With this we explain the symmetry in $R$ and $L$ in the results often observed in the literature and show that the maximal frequency spectrum depends only on the area $A=RL$ and not on the individual values of $R$ and $L$. Moreover, we collect and extend existing results and specify the maximum possible frequency spectrum of a QNN with an arbitrary number of layers as a function of the spectrum of its generators. In the case of arbitrary dimensional generators, where our two introduced notions of maximality differ, we extend existing Golomb ruler based results and introduce a second novel approach based on a variation of the turnpike problem, which we call the relaxed turnpike problem. We clarify comprehensively how the generators of a QNN must be chosen in order to obtain a maximal frequency spectrum for a given area $A$, thereby contributing to a deeper theoretical understanding. However, our numerical experiments show that trainability depends not only on $A = RL$, but also on the choice of $(R,L)$, so that knowledge of the maximum frequency spectrum alone is not sufficient to ensure good trainability.

Time series often exhibit non-ergodic behaviour that complicates forecasting and inference. This article proposes a likelihood-based approach for estimating ergodicity transformations that addresses such challenges. The method is broadly compatible with standard models, including Gaussian processes, ARMA, and GARCH. A detailed simulation study using geometric and arithmetic Brownian motion demonstrates the ability of the approach to recover known ergodicity transformations. A further case study on the large macroeconomic database FRED-QD shows that incorporating ergodicity transformations can provide meaningful improvements over conventional transformations or naive specifications in applied work.

We define two minimum distance estimators for dependent data by minimizing some approximated Maximum Mean Discrepancy distances between the true empirical distribution of observations and their assumed (parametric) model distribution. When the latter one is intractable, it is approximated by simulation, allowing to accommodate most dynamic processes with latent variables. We derive the non-asymptotic and the large sample properties of our estimators in the context of absolutely regular/beta-mixing random elements. Our simulation experiments illustrate the robustness of our procedures to model misspecification, particularly in comparison with alternative standard estimation methods.

We examine the impact of top-of-screen promotions on viewing time at ABEMA, a leading video streaming platform in Japan. To this end, we conduct a large-scale randomized controlled trial. Given the non-standard distribution of user viewing times, we estimate distributional treatment effects. Our estimation results document that spotlighting content through these promotions effectively boosts user engagement across diverse content types. Notably, promoting short content proves most effective in that it not only retains users but also motivates them to watch subsequent episodes.

Credit risk models are a critical decision-support tool for financial institutions, yet tightening data-protection rules (e.g., GDPR, CCPA) increasingly prohibit cross-border sharing of borrower data, even as these models benefit from cross-institution learning. Traditional default prediction suffers from two limitations: binary classification ignores default timing, treating early defaulters (high loss) equivalently to late defaulters (low loss), and centralized training violates emerging regulatory constraints. We propose a Federated Survival Learning framework with Bayesian Differential Privacy (FSL-BDP) that models time-to-default trajectories without centralizing sensitive data. The framework provides Bayesian (data-dependent) differential privacy (DP) guarantees while enabling institutions to jointly learn risk dynamics. Experiments on three real-world credit datasets (LendingClub, SBA, Bondora) show that federation fundamentally alters the relative effectiveness of privacy mechanisms. While classical DP performs better than Bayesian DP in centralized settings, the latter benefits substantially more from federation (+7.0\% vs +1.4\%), achieving near parity of non-private performance and outperforming classical DP in the majority of participating clients. This ranking reversal yields a key decision-support insight: privacy mechanism selection should be evaluated in the target deployment architecture, rather than centralized benchmarks. These findings provide actionable guidance for practitioners designing privacy-preserving decision support systems in regulated, multi-institutional environments.

In this paper, we propose a distributed framework for reducing the dimensionality of high-dimensional, large-scale, heterogeneous matrix-variate time series data using a factor model. The data are first partitioned column-wise (or row-wise) and allocated to node servers, where each node estimates the row (or column) loading matrix via two-dimensional tensor PCA. These local estimates are then transmitted to a central server and aggregated, followed by a final PCA step to obtain the global row (or column) loading matrix estimator. Given the estimated loading matrices, the corresponding factor matrices are subsequently computed. Unlike existing distributed approaches, our framework preserves the latent matrix structure, thereby improving computational efficiency and enhancing information utilization. We also discuss row- and column-wise clustering procedures for settings in which the group memberships are unknown. Furthermore, we extend the analysis to unit-root nonstationary matrix-variate time series. Asymptotic properties of the proposed method are derived for the diverging dimension of the data in each computing unit and the sample size $T$. Simulation results assess the computational efficiency and estimation accuracy of the proposed framework, and real data applications further validate its predictive performance.

We introduce a Markov chain Monte Carlo algorithm based on Sub-Cauchy Projection, a geometric transformation that generalizes stereographic projection by mapping Euclidean space into a spherical cap of a hyper-sphere, referred to as the complement of the dark side of the moon. We prove that our proposed method is uniformly ergodic for sub-Cauchy targets, namely targets whose tails are at most as heavy as a multidimensional Cauchy distribution, and show empirically its performance for challenging high-dimensional problems. The simplicity and broad applicability of our approach open new opportunities for Bayesian modeling and computation with heavy-tailed distributions in settings where most existing methods are unreliable.

Penalized smoothing is a standard tool in regression analysis. Classical approaches often rely on basis or kernel expansions, which constrain the estimator to a fixed span and impose smoothness assumptions that may be restrictive for discretely observed data. We introduce a class of penalized estimators that operate directly on the data grid, denoising sampled trajectories under minimal smoothness assumptions by penalizing local roughness through statistically calibrated difference operators. Some distributional and asymptotic properties of sample-based contrast statistics associated with the resulting linear smoothers are established under Hellinger differentiability of the model, without requiring Fréchet differentiability in function space. Simulation results confirm that the proposed estimators perform competitively across both smooth and locally irregular settings.

Physical activity is crucial for human health. With the increasing availability of large-scale mobile health data, strong associations have been found between physical activity and various diseases. However, accurately capturing this complex relationship is challenging, possibly because it varies across different subgroups of subjects, especially in large-scale datasets. To fill this gap, we propose a generalized heterogeneous functional method which simultaneously estimates functional effects and identifies subgroups within the generalized functional regression framework. The proposed method captures subgroup-specific functional relationships between physical activity and diseases, providing a more nuanced understanding of these associations. Additionally, we develop a pre-clustering method that enhances computational efficiency for large-scale data through a finer partition of subjects compared to true subgroups. We further introduce a testing procedure to assess whether the different subgroups exhibit distinct functional effects. In the real data application, we examine the impact of physical activity on the risk of dementia using the UK Biobank dataset, which includes over 96,433 participants. Our proposed method outperforms existing methods in future-day prediction accuracy, identifying three distinct subgroups, with detailed scientific interpretations for each subgroup. We also demonstrate the theoretical consistency of our methods. Codes implementing the proposed method are available at: https://github.com/xiaojing777/GHFM.

This study uses connected vehicle data to analyze speeding behavior on residential roads. A scalable pipeline processes trajectory data and supplements missing speed limits to generate summaries at OpenStreetMap's way ID level. The findings reveal a highly skewed distribution of both aggressive and reckless speeding. Based on a case study of Charlottesville, VA's connected vehicle data on residential roads, we found that 38% of segments had at least one instance of aggressive speeding, and 20% had at least one instance of reckless speeding. In addition, night time speeding is 27 times more prevalent than day time, and extreme violations on specific road segments highlight how severe the issue can be. Several segments rank among the top 10 for both aggressive and reckless speedings, indicating that there exist high-risk residential roads. These findings support the need for both spatial and behavioral interventions. The analysis provides a rich foundation for policy and planning, offering a valuable complement to traditional enforcement and planning tools. In conclusion, this framework sets the foundation for future applications in traffic safety analytics, demonstrating the growing potential of telematics data to inform safer, more livable communities.

Analysis of biological rhythm data often involves performing least squares trigonometric regression, which models the oscillations of a response over time as a sum of sinusoidal components. When the response is not normally distributed, an investigator will either transform the response before applying least squares trigonometric regression or extend trigonometric regression to a generalized linear model (GLM) framework. In this note, we compare these two approaches when the number of oscillation harmonics is underspecified. We assume data are sampled under an equispaced experimental design and that a log link function would be appropriate for a GLM. We show that when the response follows a generalized gamma distribution, least squares trigonometric regression with a log-transformed response, or log-normal trigonometric regression, produces unbiased parameter estimates for the oscillation harmonics, even when the number of oscillation harmonics is underspecified. In contrast, GLMs require correct specification to produce unbiased parameter estimates. We apply both methods to cortisol level data and find that only log-normal trigonometric regression produces parameter estimates that are invariant to the number of specified oscillation harmonics. Additionally, when a sufficiently large number of oscillation harmonics is specified, both methods produce identical parameter estimates for the oscillation harmonics.

Numerous offline and model-based reinforcement learning systems incorporate world models to emulate the inherent environments. A world model is particularly important in scenarios where direct interactions with the real environment is costly, dangerous, or impractical. The efficacy and interpretability of such world models are notably contingent upon the quality of the underlying training data. In this context, we introduce Action Shapley as an agnostic metric for the judicious and unbiased selection of training data. To facilitate the computation of Action Shapley, we present a randomized dynamic algorithm specifically designed to mitigate the exponential complexity inherent in traditional Shapley value computations. Through empirical validation across five data-constrained real-world case studies, the algorithm demonstrates a computational efficiency improvement exceeding 80\% in comparison to conventional exponential time computations. Furthermore, our Action Shapley-based training data selection policy consistently outperforms ad-hoc training data selection.

Differential abundance (DA) analysis in microbiome studies has recently been used to uncover a plethora of associations between microbial composition and various health conditions. While current approaches to DA typically apply only to cross-sectional data, many studies feature a longitudinal design to better understand the underlying microbial dynamics. To study DA in longitudinal microbial studies, we introduce a novel varying coefficient mixed-effects model with local sparsity. The proposed method can identify time intervals of significant group differences while accounting for temporal dependence. Specifically, we exploit a penalized kernel smoothing approach for parameter estimation and include a random effect to account for serial correlation. In particular, our method operates effectively regardless of whether sampling times are shared across subjects, accommodating irregular sampling and missing observations. Simulation studies demonstrate the necessity of modeling dependence for precise estimation and support recovery. The application of our method to a longitudinal study of mice oral microbiome during cancer development revealed significant scientific insights that were otherwise not discernible through cross-sectional analyses. An R implementation is available at https://github.com/fontaine618/LSVCMM.

In this expository article, we summarize what is known about maximum likelihood thresholds of Gaussian models, paying special attention to connections with rigidity theory.

How do evaluation systems compress multidimensional performance information into summary ratings? Using expert assessments of 9,669 professional soccer players on 28 attributes, we characterize the dimensional structure of evaluation outputs. The first principal component explains 40.6% of attribute variance, indicating a strong general factor, but formal noise discrimination procedures retain four components and bootstrap resampling confirms that this structure is highly stable. Internal consistency is high without evidence of redundancy. In out of sample prediction of expert overall ratings, a comprehensive model using the full attribute set substantially outperforms a single-factor summary (cross-validated R squared = 0.814). Overall, performance evaluations exhibit moderate information compression; they combine shared variance with stable residual dimensions that are economically meaningful for evaluation outcomes, with direct implications for the design of measurement systems.

A probabilistic formulation of irreversible kinetics is introduced in which incrementally admissible histories are weighted by a Gibbs-type measure built from an energy-dissipation action and observation constraints, with Theta controlling epistemic uncertainty. This measure can be interpreted as a Bayesian posterior over histories. In the zero-uncertainty limit, it concentrates on maximum-a-posteriori (MAP) histories, recovering classical deterministic evolution by incremental minimization in the convex generalized-standard-material setting, while allowing multiple competing MAP histories for non-convex energies or temporally coupled constraints. This emergence is demonstrated across seven distinct forward-in-time examples and an inverse inference problem of unknown histories from sparse observations via a global constrained minimum-action principle.

We consider sampling from a Gibbs distribution by evolving a finite number of particles using a particular score estimator rather than Brownian motion. To accelerate the particles, we consider a second-order score-based ODE, similar to Nesterov acceleration. In contrast to traditional kernel density score estimation, we use the recently proposed regularized Wasserstein proximal method, yielding the Accelerated Regularized Wasserstein Proximal method (ARWP). We provide a detailed analysis of continuous- and discrete-time non-asymptotic and asymptotic mixing rates for Gaussian initial and target distributions, using techniques from Euclidean acceleration and accelerated information gradients. Compared with the kinetic Langevin sampling algorithm, the proposed algorithm exhibits a higher contraction rate in the asymptotic time regime. Numerical experiments are conducted across various low-dimensional experiments, including multi-modal Gaussian mixtures and ill-conditioned Rosenbrock distributions. ARWP exhibits structured and convergent particles, accelerated discrete-time mixing, and faster tail exploration than the non-accelerated regularized Wasserstein proximal method and kinetic Langevin methods. Additionally, ARWP particles exhibit better generalization properties for some non-log-concave Bayesian neural network tasks.

Open weight models, which are ubiquitous, rarely provide access to their training data or loss function. This makes modifying such models for tasks such as pruning or unlearning, which are constrained by this unavailability, an active area of research. Existing techniques typically require gradients or ground-truth labels, rendering them infeasible in settings with limited computational resources. In this work, we investigate the fundamental question of identifying components that are critical to the model's predictive performance, without access to either gradients or the loss function, and with only distributional access such as synthetic data. We theoretically demonstrate that the global error is linearly bounded by local reconstruction errors for Lipschitz-continuous networks such as CNNs and well-trained Transformers (which, contrary to existing literature, we find exhibit Lipschitz continuity). This motivates using the locally reconstructive behavior of component subsets to quantify their global importance, via a metric that we term Subset Fidelity. In the uncorrelated features setting, selecting individual components based on their Subset Fidelity scores is optimal, which we utilize to propose ModHiFi, an algorithm for model modification that requires neither training data nor access to a loss function. ModHiFi-P, for structured pruning, achieves an 11\% speedup over the current state of the art on ImageNet models and competitive performance on language models. ModHiFi-U, for classwise unlearning, achieves complete unlearning on CIFAR-10 without fine-tuning and demonstrates competitive performance on Swin Transformers.

Transfer learning is a key component of modern machine learning, enhancing the performance of target tasks by leveraging diverse data sources. Simultaneously, overparameterized models such as the minimum-$\ell_2$-norm interpolator (MNI) in high-dimensional linear regression have garnered significant attention for their remarkable generalization capabilities, a property known as benign overfitting. Despite their individual importance, the intersection of transfer learning and MNI remains largely unexplored. Our research bridges this gap by proposing a novel two-step Transfer MNI approach and analyzing its trade-offs. We characterize its non-asymptotic excess risk and identify conditions under which it outperforms the target-only MNI. Our analysis reveals free-lunch covariate shift regimes, where leveraging heterogeneous data yields the benefit of knowledge transfer at limited cost. To operationalize our findings, we develop a data-driven procedure to detect informative sources and introduce an ensemble method incorporating multiple informative Transfer MNIs. Finite-sample experiments demonstrate the robustness of our methods to model and data heterogeneity, confirming their advantage.

Neural diffusion processes provide a scalable, non-Gaussian approach to modelling distributions over functions, but existing formulations are limited to single-task inference and do not capture dependencies across related tasks. In many multi-task regression settings, jointly modelling correlated functions and enabling task-aware conditioning is crucial for improving predictive performance and uncertainty calibration, particularly in low-data regimes. We propose multi-task neural diffusion processes, an extension that incorporates a task encoder to enable task-conditioned probabilistic regression and few-shot adaptation across related functions. The task encoder extracts a low-dimensional representation from context observations and conditions the diffusion model on this representation, allowing information sharing across tasks while preserving input-size agnosticity and the equivariance properties of neural diffusion processes. The resulting framework retains the expressiveness and scalability of neural diffusion processes while enabling efficient transfer to unseen tasks. Empirical results demonstrate improved point prediction accuracy and better-calibrated predictive uncertainty compared to single-task neural diffusion processes and Gaussian process baselines. We validate the approach on real wind farm data appropriate for wind power prediction. In this high-impact application, reliable uncertainty quantification directly supports operational decision-making in wind farm management, illustrating effective few-shot adaptation in a challenging real-world multi-task regression setting.

Many marketing applications, including credit card incentive programs, offer rewards to customers who exceed specific spending thresholds to encourage increased consumption. Quantifying the causal effect of these thresholds on customers is crucial for effective marketing strategy design. Although regression discontinuity design is a standard method for such causal inference tasks, its assumptions can be violated when customers, aware of the thresholds, strategically manipulate their spending to qualify for the rewards. To address this issue, we propose a novel framework for estimating the causal effect under threshold manipulation. The main idea is to model the observed spending distribution as a mixture of two distributions: one representing customers strategically affected by the threshold, and the other representing those unaffected. To fit the mixture model, we adopt a two-step Bayesian approach consisting of modeling non-bunching customers and fitting a mixture model to a sample around the threshold. We show posterior contraction of the resulting posterior distribution of the causal effect under large samples. Furthermore, we extend this framework to a hierarchical Bayesian setting to estimate heterogeneous causal effects across customer subgroups, allowing for stable inference even with small subgroup sample sizes. We demonstrate the effectiveness of our proposed methods through simulation studies and illustrate their practical implications using a real-world marketing dataset.

We study randomized quasi-Monte Carlo (RQMC) estimation of a multivariate integral where one of the variables takes only a finite number of values. This problem arises when the variable of integration is drawn from a mixture distribution as is common in importance sampling and also arises in some recent work on transport maps. We find that when integration error decreases at an RQMC rate that it is then important to oversample the smallest mixture components instead of using a proportional allocation. This can even improve the rate of convergence. The optimal allocations depend on the possibly unknown convergence rate. Designing the sample with an incorrect assumption on the rate still attains that convergence rate, with an inferior implied constant. The penalty for using a pessimistic rate is typically higher than for using an optimistic one. We also find that for the most accurate RQMC sampling methods, it is advantageous to arrange that our $n=2^m$ randomized Sobol' points split into subsample sizes that are also powers of $2$.

We consider the following inverse problem: Suppose a $(1+1)$-dimensional wave equation on $\mathbb{R}_+$ with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. We first establish that direct problem is well-posed. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. This approach has applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination.

Meta-analysis is widely used to integrate results from multiple experiments to obtain generalized insights. Since meta-analysis datasets are often heteroscedastic due to varying subgroups and temporal heterogeneity arising from experiments conducted at different time points, the typical meta-analysis approach, which assumes homoscedasticity, fails to adequately address this heteroscedasticity among experiments. This paper proposes a new Bayesian estimation method that simultaneously shrinks estimates of the means and variances of experiments using a hierarchical Bayesian approach while accounting for time effects through a Gaussian process. This method connects experiments via the hierarchical framework, enabling "borrowing strength" between experiments to achieve high-precision estimates of each experiment's mean. The method can flexibly capture potential time trends in datasets by modeling time effects with the Gaussian process. We demonstrate the effectiveness of the proposed method through simulation studies and illustrate its practical utility using a real marketing promotions dataset.

Popular deterministic approximations of posterior distributions from, e.g. the Laplace method, variational Bayes and expectation-propagation, generally rely on symmetric approximating families, often taken to be Gaussian. This choice facilitates optimization and inference, but typically affects the quality of the overall approximation. In fact, even in basic parametric models, the posterior distribution often displays asymmetries that yield bias and a reduced accuracy when considering symmetric approximations. Recent research has moved towards more flexible approximating families which incorporate skewness. However, current solutions are often model specific, lack a general supporting theory, increase the computational complexity of the optimization problem, and do not provide a broadly applicable solution to incorporate skewness in any symmetric approximation. This article addresses such a gap by introducing a general and provably optimal strategy to perturb any off-the-shelf symmetric approximation of a generic posterior distribution. This novel perturbation scheme is derived without additional optimization steps, and yields a similarly tractable approximation within the class of skew-symmetric densities that provably enhances the finite sample accuracy of the original symmetric counterpart. Furthermore, under suitable assumptions, it improves the convergence rate to the exact posterior by at least a $\sqrt{n}$ factor, in asymptotic regimes. These advancements are illustrated in numerical studies focusing on skewed perturbations of state-of-the-art Gaussian approximations.

Variance reduction for causal inference in the presence of network interference is often achieved through either outcome modeling, typically analyzed under unit-randomized Bernoulli designs, or clustered experimental designs, typically analyzed without strong parametric assumptions. In this work, we study the intersection of these two approaches and make the following threefold contributions. First, we present an estimator of the total treatment effect (or global average treatment effect) in low-order outcome models when the data are collected under general experimental designs, generalizing previous results for Bernoulli designs. We refer to this estimator as the pseudoinverse estimator and give bounds on its bias and variance in terms of properties of the experimental design. Second, we evaluate these bounds for the case of Bernoulli graph cluster randomized (GCR) designs. Its variance scales like the smaller of the variance obtained by the estimator derived under a low-order assumption, and the variance obtained from cluster randomization, showing that combining these variance reduction strategies is preferable to using either individually. When the order of the potential outcomes model is correctly specified, our estimator is always unbiased, and under a misspecified model, we upper bound the bias by the closeness of the ground truth model to a low-order model. Third, we give empirical evidence that our variance bounds can be used to select a good clustering that minimizes the worst-case variance under a cluster randomized design from a set of candidate clusterings. Across a range of graphs and clustering algorithms, our method consistently selects clusterings that perform well on a range of response models, suggesting the practical use of our bounds.

Gene therapies aim to address the root causes of diseases, particularly those stemming from rare genetic defects that can be life-threatening or severely debilitating. Although an increasing number of gene therapies have received regulatory approvals in recent years, understanding their long-term efficacy in trials with limited follow-up time remains challenging. To address this critical question, we propose a novel Bayesian framework designed to selectively integrate relevant external data with internal trial data to improve the inference of the durability of long-term efficacy. We proved that the proposed method has desired theoretical properties, such as identifying and favoring external subsets deemed relevant, where the relevance is defined as the similarity, induced by the marginal likelihood, between the generating mechanisms of the internal data and the selected external data. We also conducted comprehensive simulations to evaluate its performance under various scenarios. Furthermore, we apply this method to predict and infer the endogenous factor IX (FIX) levels of patients who receive Etranacogene dezaparvovec over the long-term. Our estimated long-term FIX levels, validated by recent trial data, indicate that Etranacogene dezaparvovec induces sustained FIX production. Together, the theoretical findings, simulation results, and successful application of this framework underscore its potential to address similar long-term effectiveness estimation and inference questions in real world applications.

Robustness is a fundamental pillar of Machine Learning (ML) classifiers, substantially determining their reliability. Methods for assessing classifier robustness are therefore essential. In this work, we address the challenge of evaluating corruption robustness in a way that allows comparability and interpretability on a given dataset. We propose a test data augmentation method that uses a robustness distance $ε$ derived from the datasets minimal class separation distance. The resulting MSCR (minimal separation corruption robustness) metric allows a dataset-specific comparison of different classifiers with respect to their corruption robustness. The MSCR value is interpretable, as it represents the classifiers avoidable loss of accuracy due to statistical corruptions. On 2D and image data, we show that the metric reflects different levels of classifier robustness. Furthermore, we observe unexpected optima in classifiers robust accuracy through training and testing classifiers with different levels of noise. While researchers have frequently reported on a significant tradeoff on accuracy when training robust models, we strengthen the view that a tradeoff between accuracy and corruption robustness is not inherent. Our results indicate that robustness training through simple data augmentation can already slightly improve accuracy.