Papers

While generative models, especially large language models (LLMs), are ubiquitous in today's world, principled mechanisms to assess their (in)correctness are limited. Using the conformal prediction framework, previous works construct sets of LLM responses where the probability of including an incorrect response, or error, is capped at a user-defined tolerance level. However, since these methods are based on p-values, they are susceptible to p-hacking, i.e., choosing the tolerance level post-hoc can invalidate the guarantees. We therefore leverage e-values to complement generative model outputs with e-scores as measures of incorrectness. In addition to achieving the guarantees as before, e-scores further provide users with the flexibility of choosing data-dependent tolerance levels while upper bounding size distortion, a post-hoc notion of error. We experimentally demonstrate their efficacy in assessing LLM outputs under different forms of correctness: mathematical factuality and property constraints satisfaction.

Stochastic differential equations (SDEs) are well suited to modelling noisy and irregularly sampled time series found in finance, physics, and machine learning. Traditional approaches require costly numerical solvers to sample between arbitrary time points. We introduce Neural Stochastic Flows (NSFs) and their latent variants, which directly learn (latent) SDE transition laws using conditional normalising flows with architectural constraints that preserve properties inherited from stochastic flows. This enables one-shot sampling between arbitrary states and yields up to two orders of magnitude speed-ups at large time gaps. Experiments on synthetic SDE simulations and on real-world tracking and video data show that NSFs maintain distributional accuracy comparable to numerical approaches while dramatically reducing computation for arbitrary time-point sampling.

Pretrained Transformers demonstrate remarkable in-context learning (ICL) capabilities, enabling them to adapt to new tasks from demonstrations without parameter updates. However, theoretical studies often rely on simplified architectures (e.g., omitting MLPs), plain data models (e.g., linear regression with isotropic inputs), and single-source training, limiting their relevance to realistic settings. In this work, we study ICL in pretrained Transformers with nonlinear MLP heads on nonlinear tasks drawn from multiple data sources with heterogeneous input, task, and noise distributions. We analyze a model where the MLP comprises two layers, with the first layer trained via a single gradient step and the second layer fully optimized. Under high-dimensional asymptotics, we prove that such models are equivalent in ICL error to structured polynomial predictors, leveraging results from the theory of Gaussian universality and orthogonal polynomials. This equivalence reveals that nonlinear MLPs meaningfully enhance ICL performance, particularly on nonlinear tasks, compared to linear baselines. It also enables a precise analysis of data mixing effects: we identify key properties of high-quality data sources (low noise, structured covariances) and show that feature learning emerges only when the task covariance exhibits sufficient structure. These results are validated empirically across various activation functions, model sizes, and data distributions. Finally, we experiment with a real-world scenario involving multilingual sentiment analysis where each language is treated as a different source. Our experimental results for this case exemplify how our findings extend to real-world cases. Overall, our work advances the theoretical foundations of ICL in Transformers and provides actionable insight into the role of architecture and data in ICL.

In modern industrial settings, advanced acquisition systems allow for the collection of data in the form of profiles, that is, as functional relationships linking responses to explanatory variables. In this context, statistical process monitoring (SPM) aims to assess the stability of profiles over time in order to detect unexpected behavior. This review focuses on SPM methods that model profiles as functional data, i.e., smooth functions defined over a continuous domain, and apply functional data analysis (FDA) tools to address limitations of traditional monitoring techniques. A reference framework for monitoring multivariate functional data is first presented. This review then offers a focused survey of several recent FDA-based profile monitoring methods that extend this framework to address common challenges encountered in real-world applications. These include approaches that integrate additional functional covariates to enhance detection power, a robust method designed to accommodate outlying observations, a real-time monitoring technique for partially observed profiles, and two adaptive strategies that target the characteristics of the out-of-control distribution. These methods are all implemented in the R package funcharts, available on CRAN. Finally, a review of additional existing FDA-based profile monitoring methods is also presented, along with suggestions for future research.

The partial correlation graphical LASSO (PCGLASSO) is a penalised likelihood method for Gaussian graphical models which provides scale invariant sparse estimation of the precision matrix and improves upon the popular graphical LASSO method. However, the PCGLASSO suffers from computational challenges due to the non-convexity of its associated optimisation problem. This paper provides some important breakthroughs in the computation of the PCGLASSO. First, the existence of the PCGLASSO estimate is proven when the sample size is smaller than the dimension - a case in which the maximum likelihood estimate does not exist. This means that the PCGLASSO can be used with any Gaussian data. Second, a new alternating algorithm for computing the PCGLASSO is proposed and implemented in the R package PCGLASSO available at https://github.com/JackStorrorCarter/PCGLASSO. This was the first publicly available implementation of the PCGLASSO and provides competitive computation time for moderate dimension size.

Visualizing changes over time is fundamental to learning from the past and anticipating the future. However, temporal semantics can be complicated, and existing visualization tools often struggle to accurately represent these complexities. It is common to use bespoke plot helper functions designed to produce specific graphics, due to the absence of flexible general tools that respect temporal semantics. We address this problem by proposing a grammar of temporal graphics, and an associated software implementation, 'ggtime', that encodes temporal semantics into a declarative grammar for visualizing temporal data. The grammar introduces new composable elements that support visualization across linear, cyclical, quasi-cyclical, and other granularities; standardization of irregular durations; and alignment of time points across different granularities and time zones. It is designed for interoperability with other semantic variables, allowing navigation across the space of visualizations while preserving temporal semantics.

We extend a heuristic method for automatic dimensionality selection, which maximizes a profile likelihood to identify "elbows" in scree plots. Our extension enables researchers to make automatic choices of multiple hyper-parameters simultaneously. To facilitate our extension to multi-dimensions, we propose a "softened" profile likelihood. We present two distinct parameterizations of our solution and demonstrate our approach on elastic nets, support vector machines, and neural networks. We also report a small simulation study to investigate violations to an assumption we make, and briefly discuss applications of our method to other data-analytic tasks than hyper-parameter selection.

Weather predictions are often provided as ensembles generated by repeated runs of numerical weather prediction models. These forecasts typically exhibit bias and inaccurate dependence structures due to numerical and dispersion errors, requiring statistical postprocessing for improved precision. A common correction strategy is the two-step approach: first adjusting the univariate forecasts, then reconstructing the multivariate dependence. The second step is usually handled with nonparametric methods, which can underperform when historical data are limited. Parametric alternatives, such as the Gaussian Copula Approach (GCA), offer theoretical advantages but often produce poorly calibrated multivariate forecasts due to random sampling of the corrected univariate margins. In this work, we introduce COBASE, a novel copula-based postprocessing framework that preserves the flexibility of parametric modeling while mimicking the nonparametric techniques through a rank-shuffling mechanism. This design ensures calibrated margins and realistic dependence reconstruction. We evaluate COBASE on multi-site 2-meter temperature forecasts from the ALADIN-LAEF ensemble over Austria and on joint forecasts of temperature and dew point temperature from the ECMWF system in the Netherlands. Across all regions, COBASE variants consistently outperform traditional copula-based approaches, such as GCA, and achieve performance on par with state-of-the-art nonparametric methods like SimSchaake and ECC, with only minimal differences across settings. These results position COBASE as a competitive and robust alternative for multivariate ensemble postprocessing, offering a principled bridge between parametric and nonparametric dependence reconstruction.

We study the Sobolev IPM problem for probability measures supported on a graph metric space, where critic function is constrained to lie within the unit ball defined by Sobolev norm. Sobolev IPM is intrinsically coupled with $L^p$ geometric structure within its definition, limiting its ability to incorporate other prior geometry beyond the $L^p$ paradigm. Conversely, the classic optimal transport is flexible, easy to adapt to various geometric structures by simply changing its ground cost. An important example is Orlicz-Wasserstein (OW) which utilizes Orlicz geometric structure to generalize the $L^p$ within the standard $p$-order Wasserstein, and remarkably play a vital role to advance machine learning methodologies. Inspired by recent advantages of OW, in this work, we leverage a specific class of convex functions for Orlicz geometry to mitigate such limitation for Sobolev IPM, and propose the generalized Sobolev IPM (GSI). Our GSI approach encompasses Sobolev IPM as a special case while accommodating diverse geometric priors beyond $L^p$. It however brings up significant computational hurdles that compound those already notoriously inherent in Sobolev IPM. To address these challenges, we theoretically establish a novel connection between Orlicz-Sobolev norm and Musielak norm which facilitates a novel efficient regularization for GSI. By further exploiting the underlying graph structure, we show that the regularized GSI reduces to a simple univariate optimization problem, achieving notably computational efficiency, enabling its usage in practical applications. We empirically illustrate that the regularized GSI is several-order faster than the popular OW in computation, and its performances compare favorably to other transport baselines for comparing graph-based measures in document classification and topological data analysis.

Recent advancements in technology have established terrestrial laser scanners (TLS) as a powerful instrument in geodetic deformation analysis. As TLS becomes increasingly integrated into this field, it is essential to develop a comprehensive stochastic model that accurately captures the measurement uncertainties. A key component of this model is the construction of a complete and valid variance-covariance matrix (VCM) for TLS polar measurements, which requires the estimation of variances for range, vertical, and horizontal angles, as well as their correlations. While angular variances can be obtained from manufacturer specifications, the range variance varies with different intensity measurements. As a primary contribution, this study presents an effective methodology for measuring and estimating TLS range variances using both raw and scaled intensity values. A two-dimensional scanning approach is applied to both controlled targets and arbitrary objects using TLS instruments that provide raw intensity values (e.g., Z+F~Imager~5016A) and those that output scaled intensities (e.g., Leica~ScanStation~P50). The methodology is further evaluated using field observations on a water dam surface. Overall, this work introduces a comprehensive workflow for modeling range uncertainties in high-end TLS systems.

Machine learning approaches for image classification have led to impressive advances in that field. For example, convolutional neural networks are able to achieve remarkable image classification accuracy across a wide range of applications in industry, defense, and other areas. While these machine learning models boast impressive accuracy, a related concern is how to assess and maintain calibration in the predictions these models make. A classification model is said to be well calibrated if its predicted probabilities correspond with the rates events actually occur. While there are many available methods to assess machine learning calibration and recalibrate faulty predictions, less effort has been spent on developing approaches that continually monitor predictive models for potential loss of calibration as time passes. We propose a cumulative sum-based approach with dynamic limits that enable detection of miscalibration in both traditional process monitoring and concept drift applications. This enables early detection of operational context changes that impact image classification performance in the field. The proposed chart can be used broadly in any situation where the user needs to monitor probability predictions over time for potential lapses in calibration. Importantly, our method operates on probability predictions and event outcomes and does not require under-the-hood access to the machine learning model.

Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and quantization. The spectral-norm robustness of low-rank inverse approximations remains poorly understood. We systematically study the spectral-norm error $\| (\tilde{A}^{-1})_p - A_p^{-1} \|$ for an $n\times n$ symmetric matrix $A$, where $A_p^{-1}$ denotes the best rank-\(p\) approximation of $A^{-1}$, and $\tilde{A} = A + E$ is a noisy observation. Under mild assumptions on the noise, we derive sharp non-asymptotic perturbation bounds that reveal how the error scales with the eigengap, spectral decay, and noise alignment with low-curvature directions of $A$. Our analysis introduces a novel application of contour integral techniques to the non-entire function $f(z) = 1/z$, yielding bounds that improve over naive adaptations of classical full-inverse bounds by up to a factor of $\sqrt{n}$. Empirically, our bounds closely track the true perturbation error across a variety of real-world and synthetic matrices, while estimates based on classical results tend to significantly overpredict. These findings offer practical, spectrum-aware guarantees for low-rank inverse approximations in noisy computational environments.

We propose a method for variable selection in the intensity function of spatial point processes that combines sparsity-promoting estimation with noise-robust model selection. As high-resolution spatial data becomes increasingly available through remote sensing and automated image analysis, identifying spatial covariates that influence the localization of events is crucial to understand the underlying mechanism. However, results from automated acquisition techniques are often noisy, for example due to measurement uncertainties or detection errors, which leads to spurious displacements and missed events. We study the impact of such noise on sparse point-process estimation across different models, including Poisson and Thomas processes. To improve noise robustness, we propose to use stability selection based on point-process subsampling and to incorporate a non-convex best-subset penalty to enhance model-selection performance. In extensive simulations, we demonstrate that such an approach reliably recovers true covariates under diverse noise scenarios and improves both selection accuracy and stability. We then apply the proposed method to a forestry data set, analyzing the distribution of trees in relation to elevation and soil nutrients in a tropical rain forest. This shows the practical utility of the method, which provides a systematic framework for robust variable selection in spatial point-process models under noise, without requiring additional knowledge of the process.

Recently proposed generative models for discrete data, such as Masked Diffusion Models (MDMs), exploit conditional independence approximations to reduce the computational cost of popular Auto-Regressive Models (ARMs), at the price of some bias in the sampling distribution. We study the resulting computation-vs-accuracy trade-off, providing general error bounds (in relative entropy) that depend only on the average number of tokens generated per iteration and are independent of the data dimensionality (i.e. sequence length), thus supporting the empirical success of MDMs. We then investigate the gain obtained by using non-constant schedule sizes (i.e. varying the number of unmasked tokens during the generation process) and identify the optimal schedule as a function of a so-called information profile of the data distribution, thus allowing for a principled optimization of schedule sizes. We define methods directly as sampling algorithms and do not use classical derivations as time-reversed diffusion processes, leading us to simple and transparent proofs.

Many rare diseases offer limited established treatment options, leading patients to switch therapies when new medications emerge. To analyze the impact of such treatment switches within the low sample size limitations of rare disease trials, it is important to use all available data sources. This, however, is complicated when usage of measurement instruments change during the observation period, for example when instruments are adapted to specific age ranges. The resulting disjoint longitudinal data trajectories, complicate the application of traditional modeling approaches like mixed-effects regression. We tackle this by mapping observations of each instrument to a aligned low-dimensional temporal trajectory, enabling longitudinal modeling across instruments. Specifically, we employ a set of variational autoencoder architectures to embed item values into a shared latent space for each time point. Temporal disease dynamics and treatment switch effects are then captured through a mixed-effects regression model applied to latent representations. To enable statistical inference, we present a novel statistical testing approach that accounts for the joint parameter estimation of mixed-effects regression and variational autoencoders. The methodology is applied to quantify the impact of treatment switches for patients with spinal muscular atrophy. Here, our approach aligns motor performance items from different measurement instruments for mixed-effects regression and maps estimated effects back to the observed item level to quantify the treatment switch effect. Our approach allows for model selection as well as for assessing effects of treatment switching. The results highlight the potential of modeling in joint latent representations for addressing small data challenges.

We consider the problem of density estimation in the context of multiscale Langevin diffusion processes, where a single-scale homogenized surrogate model can be derived. In particular, our aim is to learn the density of the invariant measure of the homogenized dynamics from a continuous-time trajectory generated by the full multiscale system. We propose a spectral method based on a truncated Fourier expansion with Hermite functions as orthonormal basis. The Fourier coefficients are computed directly from the data owing to the ergodic theorem. We prove that the resulting density estimator is robust and converges to the invariant density of the homogenized model as the scale separation parameter vanishes, provided the time horizon and the number of Fourier modes are suitably chosen in relation to the multiscale parameter. The accuracy and reliability of this methodology is further demonstrated through a series of numerical experiments.

We study the convergence of off-policy TD(0) with linear function approximation when used to approximate the expected discounted reward in a Markov chain. It is well known that the combination of off-policy learning and function approximation can lead to divergence of the algorithm. Existing results for this setting modify the algorithm, for instance by reweighing the updates using importance sampling. This establishes convergence at the expense of additional complexity. In contrast, our approach is to analyse the standard algorithm, but to restrict our attention to the class of reversible Markov chains. We demonstrate convergence under this mild reversibility condition on the structure of the chain, which in many applications can be assumed using domain knowledge. In particular, we establish a convergence guarantee under an upper bound on the discount factor in terms of the difference between the on-policy and off-policy process. This improves upon known results in the literature that state that convergence holds for a sufficiently small discount factor by establishing an explicit bound. Convergence is with probability one and achieves projected Bellman error equal to zero. To obtain these results, we adapt the stochastic approximation framework that was used by Tsitsiklis and Van Roy [1997 for the on-policy case, to the off-policy case. We illustrate our results using different types of reversible Markov chains, such as one-dimensional random walks and random walks on a weighted graph.

We propose a function-valued evaluation metric for generative models based on the relative density ratio (RDR) designed to characterize distributional differences between real and generated samples. As an evaluation metric, the RDR function preserves $\phi$-divergence between two distributions, enables sample-level evaluation that facilitates downstream investigations of feature-specific distributional differences, and has a bounded range that affords clear interpretability and numerical stability. Function estimation of the RDR is achieved efficiently through optimization on the variational form of $\phi$-divergence. We provide theoretical convergence rate guarantees for general estimators based on M-estimator theory, as well as the convergence rate of neural network-based estimators when the true ratio is in the anisotropic Besov space. We demonstrate the power of the proposed RDR-based evaluation through numerical experiments on MNIST, CelebA64, and the American Gut project microbiome data. We show that the estimated RDR enables not only effective overall comparison of competing generative models, but also a convenient way to reveal the underlying nature of goodness-of-fit. This enables one to assess support overlap, coverage, and fidelity while pinpointing regions of the sample space where generators concentrate and revealing the features that drive the most salient distributional differences.

Foundation models for zero-shot time series forecasting face challenges in efficient long-horizon prediction and reproducibility, with existing synthetic-only approaches underperforming on challenging benchmarks. This paper presents TempoPFN, a univariate time series foundation model based on linear Recurrent Neural Networks (RNNs) pre-trained exclusively on synthetic data. The model uses a GatedDeltaProduct architecture with state-weaving for fully parallelizable training across sequence lengths, eliminating the need for windowing or summarization techniques while maintaining robust temporal state-tracking. Our comprehensive synthetic data pipeline unifies diverse generators, including stochastic differential equations, Gaussian processes, and audio synthesis, with novel augmentations. In zero-shot evaluations on the Gift-Eval, fev-bench and Chronos-ZS benchmarks, TempoPFN achieves top-tier competitive performance, surpassing the majority of models trained on real-world data, while leveraging fully parallelizable training and inference. We open-source our complete data generation pipeline and training code, providing a reproducible foundation for future research.

Ensuring that classifiers are well-calibrated, i.e., their predictions align with observed frequencies, is a minimal and fundamental requirement for classifiers to be viewed as trustworthy. Existing methods for assessing multiclass calibration often focus on specific aspects associated with prediction (e.g., top-class confidence, class-wise calibration) or utilize computationally challenging variational formulations. In this work, we study scalable evaluation of multiclass calibration. To this end, we propose utility calibration, a general framework that measures the calibration error relative to a specific utility function that encapsulates the goals or decision criteria relevant to the end user. We demonstrate how this framework can unify and re-interpret several existing calibration metrics, particularly allowing for more robust versions of the top-class and class-wise calibration metrics, and, going beyond such binarized approaches, toward assessing calibration for richer classes of downstream utilities.

The COVID-19 pandemic reshaped human mobility through policy interventions and voluntary behavioral changes. Mobility adaptions helped mitigate pandemic spread, however our knowledge which environmental, social, and demographic factors helped mobility reduction and pandemic mitigation is patchy. We introduce a Bayesian hierarchical model to quantify heterogeneity in mobility responses across time and space in Germany's 400 districts using anonymized mobile phone data. Decomposing mobility into a disease-responsive component and disease-independent factors (temperature, school vacations, public holidays) allows us to quantify the impact of each factor. We find significant differences in reaction to disease spread along the urban-rural gradient, with large cities reducing mobility most strongly. Employment sectors further help explain variance in reaction strength during the first wave, while political variables gain significance during the second wave. However, reduced mobility only partially translates to lower peak incidence, indicating the influence of other hidden factors. Our results identify key drivers of mobility reductions and demonstrate that mobility behavior can serve as an operational proxy for population response.

The empirical Orlicz norm based on a random sample is defined as a natural estimator of the Orlicz norm of a univariate probability distribution. A law of large numbers is derived under minimal assumptions. The latter extends readily to a linear and a nonparametric regression model. Secondly, sufficient conditions for a central limit theorem with a standard rate of convergence are supplied. The conditions for the CLT exclude certain canonical examples, such as the empirical sub-Gaussian norm of normally distributed random variables. For the latter, we discover a nonstandard rate of $n^{1/4} \log(n)^{3/8}$, with a heavy-tailed, stable limit distribution. It is shown that in general, the empirical Orlicz norm does not admit any uniform rate of convergence for the class of distributions with bounded Orlicz norm.

We investigate the high-probability estimation of discrete distributions from an \iid sample under $\chi^2$-divergence loss. Although the minimax risk in expectation is well understood, its high-probability counterpart remains largely unexplored. We provide sharp upper and lower bounds for the classical Laplace estimator, showing that it achieves optimal performance among estimators that do not rely on the confidence level. We further characterize the minimax high-probability risk for any estimator and demonstrate that it can be attained through a simple smoothing strategy. Our analysis highlights an intrinsic separation between asymptotic and non-asymptotic guarantees, with the latter suffering from an unavoidable overhead. This work sharpens existing guarantees and advances the theoretical understanding of divergence-based estimation.

We develop a scalable class of models for latent variable estimation using composite Gaussian processes, with a focus on derivative Gaussian processes. We jointly model multiple data sources as outputs to improve the accuracy of latent variable inference under a single probabilistic framework. Similarly specified exact Gaussian processes scale poorly with large datasets. To overcome this, we extend the recently developed Hilbert space approximation methods for Gaussian processes to obtain a reduced-rank representation of the composite covariance function through its spectral decomposition. Specifically, we derive and analyze the spectral decomposition of derivative covariance functions and further study their properties theoretically. Using these spectral decompositions, our methods easily scale up to data scenarios involving thousands of samples. We validate our methods in terms of latent variable estimation accuracy, uncertainty calibration, and inference speed across diverse simulation scenarios. Finally, using a real world case study from single-cell biology, we demonstrate the potential of our models in estimating latent cellular ordering given gene expression levels, thus enhancing our understanding of the underlying biological process.

Hypergraphs provide a natural framework for modeling multiway interactions. We analyze a class of variational semi-supervised learning problems posed on random geometric hypergraphs and establish asymptotic consistency in the large-data limit. In particular, we identify scaling regimes that ensure well-posedness–yielding nontrivial label propagation rather than collapse to a constant labeling–and show that discrete minimizers converge, in the continuum, to solutions of a density-weighted p-Laplacian equation. We also propose Higher-Order Hypergraph Learning (HOHL), a multiscale regularization scheme based on powers of Laplacians associated with hypergraph-induced subgraphs. For geometric point clouds, we analyze an efficient multiscale Laplacian surrogate for HOHL and prove convergence to a higher-order Sobolev-type seminorm. Numerical experiments on standard benchmarks support the practical utility of the resulting higher-order regularization.

This paper introduces a novel spectral M-estimator, called the asymmetric Huber periodogram (AHP), as a generalization of the ordinary periodogram (PG), the quantile periodogram (QP), and the Huber periodogram (HP). The AHP is constructed via trigonometric asymmetric Huber regression (AHR), in which a specially designed loss function replaces the squared $\ell_2$ loss used to define the PG. Relative to the QP, the AHP can be more computationally efficient, and relative to the HP and PG, it provides a more comprehensive characterization by examining the data across the range of the asymmetry parameter. We establish the theoretical properties of the AHP and investigate its relationship with the corresponding asymmetric Huber spectrum (AHS). Building on the asymptotic theory, we develop confidence intervals (CIs) for the AHS and propose a Fisher-type test. Simulation studies and two applications further demonstrate the AHP's effectiveness in detecting hidden periodicities, robustness to outliers, and utility for time series clustering.

We introduce adaptive, tuning-free step size schedules for gradient-based sampling algorithms obtained as time-discretizations of Wasserstein gradient flows. The result is a suite of tuning-free sampling algorithms, including tuning-free variants of the unadjusted Langevin algorithm (ULA), stochastic gradient Langevin dynamics (SGLD), mean-field Langevin dynamics (MFLD), Stein variational gradient descent (SVGD), and variational gradient descent (VGD). More widely, our approach yields tuning-free algorithms for solving a broad class of stochastic optimization problems over the space of probability measures. Under mild assumptions (e.g., geodesic convexity and locally bounded stochastic gradients), we establish strong theoretical guarantees for our approach. In particular, we recover the convergence rate of optimally tuned versions of these algorithms up to logarithmic factors, in both nonsmooth and smooth settings. We then benchmark the performance of our methods against comparable existing approaches. Across a variety of tasks, our algorithms achieve similar performance to the optimal performance of existing algorithms, with no need to tune a step size parameter.

Vaccine randomized trials are typically designed to be blinded, ensuring that the estimated vaccine efficacy (VE) reflects the immunological effect of the vaccine. When blinding is broken, however, the estimated VE reflects not only the immunological effect but also behavioral effects stemming from participants' awareness of their treatment status. Recent work has proposed alternative causal estimands to the standard VE to address this issue, but their point identification results require a strong assumption: the absence of unmeasured common causes of infection risk and participants' belief about whether they received the vaccine. Personality traits, for example, may plausibly violate this assumption. We relax this assumption and derive nonparametric causal bounds for different types of VE. We construct these bounds using two approaches: linear programming-based and monotonicity-based methods. We further consider several possible causal structures for vaccine trials and show how the nonparametric bounds differ across these scenarios. Finally, we illustrate the performance of the proposed bounds using fully synthetic data and a semi-synthetic data example based on a COVID-19 vaccine trial.

We investigate the problem of detecting correlation between two Erdős-Rényi graphs $G(n,p)$, formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative hypothesis, they are correlated through a latent bijective mapping between their vertex sets. We develop a polynomial-time test by counting bounded degree motifs and prove its effectiveness for any constant correlation coefficient $\rho$ when the edge connecting probability satisfies $p\ge n^{-1+\delta}$ for some constant $\delta>0$. In particular, our guarantee improves the constrain of motif-counting methods from $\rho\ge \sqrt{\alpha}$ to any constant $\rho = \Omega(1)$, where $\alpha\approx 0.338$ is the Otter's constant.

We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $\mu$ is transported to a discrete target measure $\nu$, with particular attention to the OT map approximation. In this setting, Stochastic Gradient Descent (SGD) based solvers have demonstrated strong empirical performance in recent machine learning applications, yet their theoretical guarantee to approximate the OT map is an open question. In this work, we answer it positively by providing both computational and statistical convergence guarantees of SGD. Specifically, we show that SGD methods can estimate the OT map with a minimax convergence rate of $\mathcal{O}(1/\sqrt{n})$, where $n$ is the number of samples drawn from $\mu$. To establish this result, we study the averaged projected SGD algorithm, and identify a suitable projection set that contains a minimizer of the objective, even when the source measure is not compactly supported. Our analysis holds under mild assumptions on the source measure and applies to MTW cost functions,whic include $\|\cdot\|^p$ for $p \in (1, \infty)$. We finally provide numerical evidence for our theoretical results.