Papers

A very common task in data visualization is to plot many data points with some measured y-value as a function of fixed x-values. Uncertainties on the y-values are typically presented as vertical error bars that represent either a Frequentist confidence interval or Bayesian credible interval for each data point. Most of the time, these error bars represent a 68% confidence/credibility level, which leads to the intuition that a model fits the data reasonably well if its prediction lies within the error bars of roughly two thirds of the data points. Unfortunately, this and other intuitions no longer work when the uncertainties of the data points are correlated. If the error bars only show the square root of diagonal elements of some covariance matrix with non-negligible off-diagonal elements, we simply do not have enough information in the plot to judge whether a drawn model line agrees well with the data or not. In this paper we will demonstrate this problem and discuss ways to add more information to the plots to make it easier to judge the agreement between the data and some model prediction in the plot, as well as glean some insight where the model might be deficient. This is done by explicitly showing the contribution of the first principal component of the uncertainties, and by displaying the conditional uncertainties of all data points.

Advances in precision medicine increasingly drive methodological innovation in health research. A key development is the use of personalized prediction models (PPMs), which are fit using a similar subpopulation tailored to a specific index patient, and have been shown to outperform one-size-fits-all models, particularly in terms of model discrimination performance. We propose a generalized loss function that enables tuning of the subpopulation size used to fit a PPM. This loss function allows joint optimization of discrimination and calibration, allowing both the performance measures and their relative weights to be specified by the user. To reduce computational burden, we conducted extensive simulation studies to identify practical bounds for the grid of subpopulation sizes. Based on these results, we recommend using a lower bound of 20% and an upper bound of 70% of the entire training dataset. We apply the proposed method to both simulated and real-world datasets and demonstrate that previously observed relationships between subpopulation size and model performance are robust. Furthermore, we show that the choice of performance measures in the loss function influences the optimal subpopulation size selected. These findings support the flexible and computationally efficient implementation of PPMs in precision health research.

In this letter, we address the problem of developing quantum state tomography (QST) methods that remain valid at any time during a sequence of measurements. Specifically, the aim is to provide a rigorous quantification of the uncertainty associated with the current state estimate as data are acquired incrementally. To this end, the proposed framework augments existing QST techniques by associating current point estimates of the state with confidence sets that are guaranteed to contain the true quantum state with a user-defined probability. The methodology is grounded in recent statistical advances in anytime-valid confidence sequences. Numerical results confirm the theoretical coverage properties of the proposed anytime-valid QST.

Score-based diffusion models have recently been extended to infinite-dimensional function spaces, with uses such as inverse problems arising from partial differential equations. In the Bayesian formulation of inverse problems, the aim is to sample from a posterior distribution over functions obtained by conditioning a prior on noisy observations. While diffusion models provide expressive priors in function space, the theory of conditioning them to sample from the posterior remains open. We address this, assuming that either the prior lies in the Cameron-Martin space, or is absolutely continuous with respect to a Gaussian measure. We prove that the models can be conditioned using an infinite-dimensional extension of Doob's $h$-transform, and that the conditional score decomposes into an unconditional score and a guidance term. As the guidance term is intractable, we propose a simulation-free score matching objective (called Supervised Guidance Training) enabling efficient and stable posterior sampling. We illustrate the theory with numerical examples on Bayesian inverse problems in function spaces. In summary, our work offers the first function-space method for fine-tuning trained diffusion models to accurately sample from a posterior.

Biased gradient compression with error feedback (EF) reduces communication in federated learning (FL), but under non-IID data, the residual error can decay slowly, causing gradient mismatch and stalled progress in the early rounds. We propose step-ahead partial error feedback (SA-PEF), which integrates step-ahead (SA) correction with partial error feedback (PEF). SA-PEF recovers EF when the step-ahead coefficient $\alpha=0$ and step-ahead EF (SAEF) when $\alpha=1$. For non-convex objectives and $\delta$-contractive compressors, we establish a second-moment bound and a residual recursion that guarantee convergence to stationarity under heterogeneous data and partial client participation. The resulting rates match standard non-convex Fed-SGD guarantees up to constant factors, achieving $O((\eta,\eta_0TR)^{-1})$ convergence to a variance/heterogeneity floor with a fixed inner step size. Our analysis reveals a step-ahead-controlled residual contraction $\rho_r$ that explains the observed acceleration in the early training phase. To balance SAEF's rapid warm-up with EF's long-term stability, we select $\alpha$ near its theory-predicted optimum. Experiments across diverse architectures and datasets show that SA-PEF consistently reaches target accuracy faster than EF.

Sequential nested trial (SNT) emulation is a powerful approach for maximizing precision and avoiding time-related biases. However, there exists little discussion about the implied causal estimands in comparison to a real-world single point trial. We used Monte Carlo simulation to compare treatment effect estimates from an SNT emulation that re-indexed patients annually and a SNT emulation with a treatment decision design to the estimates from a single point trial. We generated 5,000 cohorts of 5,000 people with 3 years of follow-up. For the single point trial, patients were randomized to initiate or not initiate treatment at Visit 1. For the SNT emulations, simulated patients could contribute up to two index dates. When disease severity did not modify the treatment effect, both SNT approaches returned treatment effect estimates identical to the single point trial. In the presence of treatment effect modification by disease severity, both SNT approaches returned treatment effect estimates that diverged from the single point trial even after confounding-adjustment. These findings underscore the difficulties of interpreting causal estimands from a SNT emulation: the target population does not correspond to a single time point trial. Such implications are important for communicating study results for evidence-based decision-making.

Dose optimization is a hallmark of Project Optimus for oncology drug development. The number of doses to include in a dose optimization study depends on the totality of evidence, which is often unclear in early-phase development. With equal sample sizes per dose, carrying three doses is clearly more advantageous than two for optimization. In this paper, we show that, even when the total sample size is fixed, it is still preferable to carry three unless there is very strong evidence that one can be dropped. A mathematical approximation is applied to guide the investigation, followed by a simulation study to complement the theoretical findings. Semi-quantitative guidance is provided for practitioners, addressing both randomized and non-randomized dose optimization while considering population homogeneity.

We present a propagation model showing that a transmitter randomly positioned in space generates unbounded peaks in the histogram of the resulting power, provided the signal strength is an oscillating or non-monotonic function of distance. Specifically, these peaks are singularities in the empirical probability density that occur at turning point values of the deterministic propagation model. We explain the underlying mechanism of this phenomenon through a concise mathematical argument. This observation has direct implications for estimating random propagation effects such as fading, particularly when reflections off walls are involved. Motivated by understanding intelligent surfaces, we apply this fundamental result to a physical model consisting of a single transmitter between two parallel passive walls. We analyze signal fading due to reflections and observe power oscillations resulting from wall reflections – a phenomenon long studied in waveguides but relatively unexplored in wireless networks. For the special case where the transmitter is placed halfway between the walls, we present a compact closed-form expression for the received signal involving the Lerch transcendent function. The insights from this work can inform design decisions for intelligent surfaces deployed in cities.

The mean-variance model remains the most prevalent investment framework, built on diversification principles. However, it consistently struggles with estimation errors in expected returns and the covariance matrix, its core parameters. To address this concern, this research evaluates the performance of mean variance (MV) and global minimum-variance (GMV) models across various shrinkage estimators designed to improve these parameters. Specifically, we examine five shrinkage estimators for expected returns and eleven for the covariance matrix. To compare multiple portfolios, we employ a super efficient data envelopment analysis model to rank the portfolios according to investors risk-return preferences. Our comprehensive empirical investigation utilizes six real world datasets with different dimensional characteristics, applying a rolling window methodology across three out of sample testing periods. Following the ranking process, we examine the chosen shrinkage based MV or GMV portfolios against five traditional portfolio optimization techniques classical MV and GMV for sample estimates, MiniMax, conditional value at risk, and semi mean absolute deviation risk measures. Our empirical findings reveal that, in most scenarios, the GMV model combined with the Ledoit Wolf two parameter shrinkage covariance estimator (COV2) represents the optimal selection for a broad spectrum of investors. Meanwhile, the MV model utilizing COV2 alongside the sample mean (SM) proves more suitable for return oriented investors. These two identified models demonstrate superior performance compared to traditional benchmark approaches. Overall, this study lays the groundwork for a more comprehensive understanding of how specific shrinkage models perform across diverse investor profiles and market setups.

Cluster analysis relates to the task of assigning objects into groups which ideally present some desirable characteristics. When a cluster structure is confined to a subset of the feature space, traditional clustering techniques face unprecedented challenges. We present an information theoretic framework that overcomes the problems associated with sparse data, allowing for joint feature weighting and clustering. Our proposal constitutes a competitive alternative to existing clustering algorithms for sparse data, as demonstrated through simulations on synthetic data. The effectiveness of our method is established by an application on a real-world genomics data set.

Understanding the causal effects of organ-specific features from medical imaging on clinical outcomes is essential for biomedical research and patient care. We propose a novel Functional Linear Structural Equation Model (FLSEM) to capture the relationships among clinical outcomes, functional imaging exposures, and scalar covariates like genetics, sex, and age. Traditional methods struggle with the infinite-dimensional nature of exposures and complex covariates. Our FLSEM overcomes these challenges by establishing identifiable conditions using scalar instrumental variables. We develop the Functional Group Support Detection and Root Finding (FGS-DAR) algorithm for efficient variable selection, supported by rigorous theoretical guarantees, including selection consistency and accurate parameter estimation. We further propose a test statistic to test the nullity of the functional coefficient, establishing its null limit distribution. Our approach is validated through extensive simulations and applied to UK Biobank data, demonstrating robust performance in detecting causal relationships from medical imaging.

Predicting the human burden of vector-borne diseases from limited surveillance data remains a major challenge, particularly in the presence of nonlinear transmission dynamics and delayed effects arising from vector ecology and human behavior. We develop a data-driven framework based on an extension of Sparse Identification of Nonlinear Dynamics (SINDy) to systems with distributed memory, enabling discovery of transmission mechanisms directly from time series data. Using severe fever with thrombocytopenia syndrome (SFTS) as a case study, we show that this approach can uncover key features of tick-borne disease dynamics using only human incidence and local temperature data, without imposing predefined assumptions on human case reporting. We further demonstrate that predictive performance is substantially enhanced when the data-driven model is coupled with mechanistic representations of tick-host transmission pathways informed by empirical studies. The framework supports systematic sensitivity analysis of memory kernels and behavioral parameters, identifying those most influential for prediction accuracy. Although the approach prioritizes predictive accuracy over mechanistic transparency, it yields sparse, interpretable integral representations suitable for epidemiological forecasting. This hybrid methodology provides a scalable strategy for forecasting vector-borne disease risk and informing public health decision-making under data limitations.

Learning the dependence structure among variables in complex systems is a central problem across medical, natural, and social sciences. These structures can be naturally represented by graphs, and the task of inferring such graphs from data is known as graph learning or causal discovery. Existing approaches typically rely on restrictive assumptions about the data-generating process, employ greedy oracle algorithms, or solve approximate formulations of the graph learning problem. Therefore, they are either sensitive to violations of central assumptions or fail to guarantee globally optimal solutions. We address these limitations by introducing a nonparametric graph learning framework based on conditional independence testing and integer programming. We reformulate the graph learning problem as a mixed-integer program and prove that solving this integer-programming problem provides a globally optimal solution to the original graph learning problem. Our method leverages efficient encodings of graphical separation criteria, enabling the exact recovery of larger graphs than was previously feasible. We provide an open-source R package 'glip' which supports learning (acyclic) directed (mixed) graphs and chain graphs. We demonstrate that our approach is often faster than existing exact graph learning procedures and achieves state-of-the-art performance on simulated and benchmark data across all aforementioned classes of graphs.

Asynchronous primal-dual methods for decentralized non-smooth convex optimization often require each node to maintain $\mathcal{O}(d)$ auxiliary variables, where $d$ is its degree. This dependence on degree increases memory requirements and can amplify the effects of stale information, especially in dense networks. Motivated by the challenge of frugal memory management in decentralized learning, we introduce Goal-PD, an asynchronous gossip-based primal-dual algorithm that maintains only two variables per node, regardless of the node's degree. We establish almost-sure convergence of Goal-PD to a minimizer of the underlying optimization problem, and prove linear convergence when the objective functions are piecewise linear-quadratic. For decentralized mean estimation, we show that pairwise averaging is a special case of Goal-PD, which establishes a direct link between the proposed primal-dual framework and classical gossip. Experiments on synthetic and real datasets over various network topologies, with non-smooth objectives including median estimation, show that Goal-PD converges faster than existing asynchronous baselines while requiring significantly less memory by design.

Survival analysis has become a standard approach for modelling time to default by time-varying covariates in credit risk. Unlike most existing methods that implicitly assume a stationary data-generating process, in practise, mortgage portfolios are exposed to various forms of data drift caused by changing borrower behaviour, macroeconomic conditions, policy regimes and so on. This study investigates the impact of data drift on survival-based credit risk models and proposes a dynamic joint modelling framework to improve robustness under non-stationary environments. The proposed model integrates a longitudinal behavioural marker derived from balance dynamics with a discrete-time hazard formulation, combined with landmark one-hot encoding and isotonic calibration. Three types of data drift (sudden, incremental and recurring) are simulated and analysed on mortgage loan datasets from Freddie Mac. Experiments and corresponding evidence show that the proposed landmark-based joint model consistently outperforms classical survival models, tree-based drift-adaptive learners and gradient boosting methods in terms of discrimination and calibration across all drift scenarios, which confirms the superiority of our model design.

We develop a fully intrinsic Bayesian framework for nonparametric regression on the unit sphere based on isotropic Gaussian field priors and the harmonic structure induced by the Laplace-Beltrami operator. Under uniform random design, the regression model admits an exact diagonalization in the spherical harmonic basis, yielding a Gaussian sequence representation with frequency-dependent multiplicities. Exploiting this structure, we derive closed-form posterior distributions, optimal spectral truncation schemes, and sharp posterior contraction rates under integrated squared loss. For Gaussian priors with polynomially decaying angular power spectra, including spherical Matérn priors, we establish posterior contraction rates over Sobolev classes, which are minimax-optimal under correct prior calibration. We further show that the posterior mean admits an exact variational characterization as a geometrically intrinsic penalized least-squares estimator, equivalent to a Laplace-Beltrami smoothing spline.

We study the computational phase transition in a multi-frequency group synchronization problem, where pairwise relative measurements of group elements are observed across multiple frequency channels and corrupted by Gaussian noise. Using the framework of low-degree polynomial algorithms, we analyze the task of estimating the structured signal in such observations. We show that, assuming the low-degree heuristic, in synchronization models over the circle group $\mathsf{SO}(2)$, a simple spectral method is computationally optimal among all polynomial-time estimators when the number of frequencies satisfies $L=n^{o(1)}$. This significantly extends prior work \cite{KBK24+}, which only applied to a fixed constant number of frequencies. Together with known upper bounds on the statistical threshold \cite{PWBM18a}, our results establish the existence of a statistical-to-computational gap in this model when the number of frequencies is sufficiently large.

Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. However, extension of spectral diffusion approaches to spherical data raises nontrivial geometric and stochastic issues that are absent in the Euclidean setting. In this work, we develop a diffusion modeling framework defined directly on finite-dimensional spherical harmonic representations of real-valued functions on the sphere. We show that the spherical discrete Fourier transform maps spatial Brownian motion to a constrained Gaussian process in the frequency domain with deterministic, generally non-isotropic covariance. This induces modified forward- and reverse-time stochastic differential equations in the spectral domain. As a consequence, spatial and spectral score matching objectives are generally no longer equivalent, even in the band-limited setting. We establish a quantitative relationship between the two objectives, showing that the geometry-induced covariance of the spectral noise gives rise to a distinct, geometry-dependent inductive bias. We also derive the corresponding forward and reverse diffusion equations and characterize the induced noise covariance.

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

We study the asymptotic behavior of least-squares cross-validation bandwidth selection in kernel density estimation on the $d$-dimensional hypersphere, $d\geq 1$. We show that the exact rate of convergence with respect to the optimal bandwidth minimizing the mean integrated squared error, shown to exist under mild non-uniformity conditions, is $n^{-d/(2d+8)}$, thus approaching the $n^{-1/2}$ parametric rate as $d$ grows. This “blessing of dimensionality” in bandwidth selection offers theoretical support for utilizing the conceptually simpler cross-validation selector over plug-in techniques for larger dimensions $d$. We compare this result for bandwidth estimation on the $d$-dimensional Euclidean space through explicit expressions for the asymptotic variance functionals. Numerical experiments corroborate the speed of this convergence in an array of scenarios and dimensions, precisely illustrating the tipping dimension where cross-validation outperforms plug-in approaches.

Diffusion Map is a spectral dimensionality reduction technique which is able to uncover nonlinear submanifolds in high-dimensional data. And, it is increasingly applied across a wide range of scientific disciplines, such as biology, engineering, and social sciences. But data preprocessing, parameter settings and component selection have a significant influence on the resulting manifold, something which has not been comprehensively discussed in the literature so far. We provide a practice oriented review of the Diffusion Map technique, illustrate pitfalls and showcase a recently introduced technique for identifying the most relevant components. Our results show that the first components are not necessarily the most relevant ones.

Pearl's causal hierarchy is a foundational lens for formulating causal questions and is most often discussed within the framework of structural causal models. We recast the hierarchy in potential outcomes language and make its information ordering operational at the level of causal estimands. Specifically, we classify an estimand according to whether it requires marginal potential outcome distributions, their joint distribution or nested cross-world quantities, or individual-level counterfactual outcomes. We apply this criterion systematically to a broad range of estimands, including cases whose classification depends on the formulation of the scientific question. We then clarify what additional assumptions are needed when an estimand depends not only on the marginal distributions of potential outcomes, but also on their unobserved joint distribution. In particular, randomization identifies the marginals, whereas monotonicity, copula restrictions, rank preservation, and partial identification restrict or characterize the remaining uncertainty about the joint distribution. The resulting framework provides a practical map from a scientific question to an estimand, the probabilistic object it requires, and the assumptions needed for identification.

Randomized subspace methods reduce per-iteration cost; however, in nonconvex optimization, most analyses are expectation-based, and high-probability bounds remain scarce even under sub-Gaussian noise. We first prove that randomized subspace SGD (RS-SGD) admits a high-probability convergence bound under sub-Gaussian noise, achieving the same order of oracle complexity as prior in-expectation results. Motivated by the prevalence of heavy-tailed gradients in modern machine learning, we then propose randomized subspace normalized SGD (RS-NSGD), which integrates direction normalization into subspace updates. Assuming the noise has bounded $p$-th moments, we establish both in-expectation and high-probability convergence guarantees, and show that RS-NSGD can achieve better oracle complexity than full-dimensional normalized SGD.

We propose a unified framework to enhance the power of online multiple hypothesis testing procedures based on $e$-values. While $e$-value-based methods offer robust online False Discovery Rate (FDR) control under minimal assumptions, they often suffer from power loss by discarding evidence that exceeds the rejection threshold. We address this inefficiency via the Sequential Control with Overshoot Refund for E-values (SCORE) framework, which leverages the inequality $\mathbb{I}(y \ge 1) \le y - (y-1)_+$, valid for all $y\ge 0$, to reclaim this otherwise wasted evidence. This simple yet powerful insight yields a unified principle for improving a broad class of online testing algorithms. Building on this framework, we develop SCORE-enhanced versions of several state-of-the-art procedures, including SCORE-LOND, SCORE-LORD, and SCORE-SAFFRON, all of which strictly dominate their original counterparts while preserving valid finite-sample FDR control. Furthermore, under mild assumptions, SCORE permits retroactive updates of alpha-wealth by using the latest decision twice: first to determine its reward or loss, and then to refresh past wealth. Such a mechanism enables more aggressive testing strategies while maintaining valid FDR control, thereby further improving statistical power. The effectiveness of the proposed methods is validated through extensive simulation and real-data experiments.

In this paper, we consider a partial deconvolution kernel estimator for nonparametric regression when some covariates are measured with error while others are observed without error. We focus on a general and realistic setting in which the measurement errors are heteroscedastic. We propose a kernel-based estimator of the regression function in this framework and show that it achieves the optimal convergence rate under suitable regularity conditions. The finite-sample performance of the proposed estimator is illustrated through simulation studies.

Discovery problems often require deciding whether additional sampling is needed to detect all categories whose prevalence exceeds a prespecified threshold. We study this question under a Bernoulli product (incidence) model, where categories are observed only through presence–absence across sampling units. Our inferential target is the maximum unseen probability, the largest prevalence among categories not yet observed. We develop nonasymptotic, distribution-free upper confidence bounds for this quantity in two regimes: bounded alphabets (finite and known number of categories) and unbounded alphabets (countably infinite under a mild summability condition). We characterise the limits of data-independent worst-case bounds, showing that in the unbounded regime no nontrivial data-independent procedure can be uniformly valid. We then propose data-dependent bounds in both regimes and establish matching lower bounds demonstrating their near-optimality. We compare empirically the resulting procedures in both simulated and real datasets. Finally, we use these bounds to construct sequential stopping rules with finite-sample guarantees, and demonstrate robustness to contamination that introduces spurious low-prevalence categories.

Machine learning models in high-stakes applications, such as recidivism prediction and automated personnel selection, often exhibit systematic performance disparities across sensitive subpopulations, raising critical concerns regarding algorithmic bias. Fairness auditing addresses these risks through two primary functions: certification, which verifies adherence to fairness constraints; and flagging, which isolates specific demographic groups experiencing disparate treatment. However, existing auditing techniques are frequently limited by restrictive distributional assumptions or prohibitive computational overhead. We propose a novel empirical likelihood-based (EL) framework that constructs robust statistical measures for model performance disparities. Unlike traditional methods, our approach is non-parametric; the proposed disparity statistics follow asymptotically chi-square or mixed chi-square distributions, ensuring valid inference without assuming underlying data distributions. This framework uses a constrained optimization profile that admits stable numerical solutions, facilitating both large-scale certification and efficient subpopulation discovery. Empirically, the EL methods outperform bootstrap-based approaches, yielding coverage rates closer to nominal levels while reducing computational latency by several orders of magnitude. We demonstrate the practical utility of this framework on the COMPAS dataset, where it successfully flags intersectional biases, specifically identifying a significantly higher positive prediction rate for African-American males under 25 and a systemic under-prediction for Caucasian females relative to the population mean.

We develop unbalanced Haar (UH) wavelet tree ensembles for regression on triangulable manifolds. Given data sampled on a triangulated manifold, we construct UH wavelet trees whose atoms are supported on geodesic triangles and form an orthonormal system in $L^2(\mu_n)$, where $\mu_n$ is the empirical measure on the sample, which allows us to use UH trees as weak learners in additive ensembles. Our construction extends classical UH wavelet trees from regular Euclidean grids to generic triangulable manifolds while preserving three key properties: (i) orthogonality and exact reconstruction at the sampled locations, (ii) recursive, data-driven partitions adapted to the geometry of the manifold via geodesic triangulations, and (iii) compatibility with optimization-based and Bayesian ensemble building. In Euclidean settings, the framework reduces to standard UH wavelet tree regression and provides a baseline for comparison. We illustrate the method on synthetic regression on the sphere and on climate anomaly fields on a spherical mesh, where UH ensembles on triangulated manifolds substantially outperform classical tree ensembles and non-adaptive mesh-based wavelets. For completeness, we also report results on image denoising on regular grids. A Bayesian variant (RUHWT) provides posterior uncertainty quantification for function estimates on manifolds. Our implementation is available at http://www.github.com/hrluo/WaveletTrees.

Exhaustively evaluating many large language models (LLMs) on a large suite of benchmarks is expensive. We cast benchmarking as finite-population inference and, under a fixed query budget, seek tight confidence intervals (CIs) for model accuracy with asymptotic frequentist coverage. We propose Factorized Active Querying (FAQ), which (a) leverages historical information through a Bayesian factor model; (b) adaptively selects questions using a hybrid variance-reduction/active-learning sampling policy; and (c) maintains validity through Pro-Active Inference—a finite-population extension of active inference (Zrnic & Candès, 2024) that enables direct question selection while preserving asymptotic frequentist coverage. In our experiments, FAQ delivers up to $5\times$ effective sample size gains over strong baselines on two benchmark suites, across varying historical-data missingness levels, with negligible computational overhead: this means that it matches the CI width of uniform sampling while using up to $5\times$ fewer queries. We release our source code and our curated datasets to support reproducibility and future research.

Recently, flow-based generative models have shown superior efficiency compared to diffusion models. In this paper, we study rectified flow models, which constrain transport trajectories to be linear from the base distribution to the data distribution. This structural restriction greatly accelerates sampling, often enabling high-quality generation with a single Euler step. Under standard assumptions on the neural network classes used to parameterize the velocity field and data distribution, we prove that rectified flows achieve sample complexity $\tilde{O}(\varepsilon^{-2})$. This improves on the best known $O(\varepsilon^{-4})$ bounds for flow matching model and matches the optimal rate for mean estimation. Our analysis exploits the particular structure of rectified flows: because the model is trained with a squared loss along linear paths, the associated hypothesis class admits a sharply controlled localized Rademacher complexity. This yields the improved, order-optimal sample complexity and provides a theoretical explanation for the strong empirical performance of rectified flow models.