Papers

This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We consider matrices of the form $B_n=(1/n)T_p^{1/2}X_nX_n^{*}T_p^{1/2},$ where $X_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex random variables, and $T_p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. By using Bernstein polynomial approximation, we show that, under $\mathbb{E}|x_{ij}|^{8}<\infty$, the centered LSSs of $B_n$ have Gaussian limits. Under the stronger $\mathbb{E}|x_{ij}|^{10}<\infty$, we further establish convergence rates $O(n^{-1/2+\kappa})$ in Kolmogorov–Smirnov $O(n^{-1/2+\kappa})$, for any fixed $\kappa>0$.

Absolute anonymization, conceived as an irreversible transformation preventing re-identification and sensitive value disclosure, has proven to be a broken promise. Modern data protection must therefore shift toward a privacy-utility trade-off grounded in risk mitigation. Differential Privacy (DP) offers a rigorous mathematical framework for balancing quantified disclosure risk with analytical usefulness. Nevertheless, widespread adoption remains limited, largely because complex technical concepts, such as privacy-loss parameters, have yet to be translated into forms meaningful to non-technical stakeholders. This difficulty arises from randomization itself: both analysts and adversaries must draw conclusions from uncertain observations rather than deterministic values. In this work, we adopt an interpretation of the privacy-utility trade-off based on hypothesis testing to measure the uncertainty introduced by randomized mechanisms. In particular, we use the concept of relative disclosure risk to quantify the maximum reduction in uncertainty an adversary can obtain from a membership attack on protected outputs, and show this measure relates directly to standard privacy-loss parameters. We further analyze how DP affects analytical validity via its impact on hypothesis tests assessing statistical significance. Building on these results, we provide practical guidance, accessible to non-experts such as data protection authorities, for navigating the trade-off and selecting protection mechanisms and parameter values.

Generative modeling of three-dimensional (3D) molecules is a fundamental yet challenging problem in drug discovery and materials science. Existing approaches typically represent molecules as 3D graphs and co-generate discrete atom types with continuous atomic coordinates, leading to intrinsic learning difficulties such as heterogeneous modality entanglement and geometry-chemistry coherence constraints. We propose VecMol, a paradigm-shifting framework that reimagines molecular representation by modeling 3D molecules as continuous vector fields over Euclidean space, where vectors point toward nearby atoms and implicitly encode molecular structure. The vector field is parameterized by a neural field and generated using a latent diffusion model, avoiding explicit graph generation and decoupling structure learning from discrete atom instantiation. Experiments on the QM9 and GEOM-Drugs benchmarks validate the feasibility of this novel approach, suggesting vector-field-based representations as a promising new direction for 3D molecular generation.

This study presents a new procedure for necessary tests of multivariate normality based on the uniform distribution on the Stiefel manifold. We demonstrate that the test statistic, which is formed by the product of the scaled residual matrix and the symmetric square root of a Wishart matrix, is exactly distributed as a matrix-variate normal distribution under the null hypothesis. Monte Carlo simulations are conducted to assess the Type I error rate and power in non-asymptotic settings.

In this paper, we consider the problem of black-box optimization with noisy feedback revealed in batches, where the unknown function to optimize has a bounded norm in some Reproducing Kernel Hilbert Space (RKHS). We refer to this as the Batched Kernelized Bandits problem, and refine and extend existing results on regret bounds. For algorithmic upper bounds, (Li and Scarlett, 2022) shows that $B=O(\log\log T)$ batches suffice to attain near-optimal regret, where $T$ is the time horizon and $B$ is the number of batches. We further refine this by (i) finding the optimal number of batches including constant factors (to within $1+o(1)$), and (ii) removing a factor of $B$ in the regret bound. For algorithm-independent lower bounds, noticing that existing results only apply when the batch sizes are fixed in advance, we present novel lower bounds when the batch sizes are chosen adaptively, and show that adaptive batches have essentially same minimax regret scaling as fixed batches. Furthermore, we consider a robust setting where the goal is to choose points for which the function value remains high even after an adversarial perturbation. We present the robust-BPE algorithm, and show that a suitably-defined cumulative regret notion incurs the same bound as the non-robust setting, and derive a simple regret bound significantly below that of previous work.

Sparse regularization plays a central role in solving inverse problems arising from incomplete or corrupted measurements. Different regularizers correspond to different prior assumptions about the structure of the unknown signal, and reconstruction performance depends on how well these priors match the intrinsic sparsity of the data. This work investigates the effect of sparsity priors in inverse problems by comparing conventional L1 regularization with the Variational Garrote (VG), a probabilistic method that approximates L0 sparsity through variational binary gating variables. A unified experimental framework is constructed across multiple reconstruction tasks including signal resampling, signal denoising, and sparse-view computed tomography. To enable consistent comparison across models with different parameterizations, regularization strength is swept across wide ranges and reconstruction behavior is analyzed through train-generalization error curves. Experiments reveal characteristic bias-variance tradeoff patterns across tasks and demonstrate that VG frequently achieves lower minimum generalization error and improved stability in strongly underdetermined regimes where accurate support recovery is critical. These results suggest that sparsity priors closer to spike-and-slab structure can provide advantages when the underlying coefficient distribution is strongly sparse. The study highlights the importance of prior-data alignment in sparse inverse problems and provides empirical insights into the behavior of variational L0-type methods across different information bottlenecks.

This paper investigates change-point of variance in panel data models with time series of $\alpha$-mixing. Based on the cumulative sum (CUSUM) method and the individual differences, we construct a CUSUM test for panel data models to detect variance changes. Under the null hypothesis, we derive the limit distribution of this test, which can be used to detect the change-point of variance. Under the alternative hypothesis, the limit behavior of the CUSUM test is also derived. To validate the performance of the test, we conducted simulation analyses on with Gaussian and Gamma errors. The results demonstrate that this testing method significantly outperforms existing approaches, particularly in detecting sparse variance changes. Finally, we conducted a practical case study using panel data from the Shanghai Shenzhen CSI 300 Index Components. Not only did we successfully identify the change-points of variance, but we also delved deeper into the underlying economic drivers behind these changes.

The InfoNCE loss in contrastive learning depends critically on a temperature parameter, yet its dynamics under fixed versus annealed schedules remain poorly understood. We provide a theoretical analysis by modeling embedding evolution under Langevin dynamics on a compact Riemannian manifold. Under mild smoothness and energy-barrier assumptions, we show that classical simulated annealing guarantees extend to this setting: slow logarithmic inverse-temperature schedules ensure convergence in probability to a set of globally optimal representations, while faster schedules risk becoming trapped in suboptimal minima. Our results establish a link between contrastive learning and simulated annealing, providing a principled basis for understanding and tuning temperature schedules.

Random sample consensus (RANSAC), which is based on a repetitive sampling from a given dataset, is one of the most popular robust estimation methods. In this study, an energy-based model (EBM) for robust estimation that has a similar scheme to RANSAC, energy-based RANSAC (EB-RANSAC), is proposed. EB-RANSAC is applicable to a wide range of estimation problems similar to RANSAC. However, unlike RANSAC, EB-RANSAC does not require a troublesome sampling procedure and has only one hyperparameter. The effectiveness of EB-RANSAC is numerically demonstrated in two applications: a linear regression and maximum likelihood estimation.

We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.

Asymptotic inference using functional principal component regression (FPCR) has long been considered difficult, largely because, upon any scalar scaling, the FPCR estimator fails to satisfy a central limit theorem, leading to the prevailing belief that it is unsuitable for direct statistical inference. In this paper, we revisit this traditional viewpoint by establishing a new result: upon suitable operator scaling, valid Gaussian and bootstrap approximations hold for the FPCR estimator. We apply this surprising finding to hypothesis testing for the significance of the slope function in functional regression models and demonstrate the strong numerical performance of the resulting tests. Our finding also provides a new framework for projection inference. While concise, our results yield powerful inferential tools for functional regression. We believe it paves the way for new lines of inferential methodology for more complex functional regression settings.

We consider minimising a spectral risk objective, the expected cost plus a weighted conditional value-at-risk, when the distribution of the uncertain parameters depends on the decision and can only be sampled through a stochastic simulator under a fixed budget of draws. We propose adaptive conditional residual sampling (ACRS). It takes one simulator draw at each point of a space-filling design, fits a heteroscedastic regression model of the decision-dependent distribution and reuses the standardised residual vectors as scenarios to construct a deterministic surrogate. ACRS searches this surrogate for several local optima, adds draws around the most promising ones, ranks the candidates on the simulator with common random numbers and finishes with sample-path local refinement using tail importance sampling. We show that the optimality gap of the surrogate minimiser is bounded by the Wasserstein error of the scenario distribution and that the refinement estimator is consistent. In the experiments, all methods share the budget accounting, the design sampler and the local refinement routine, and solutions are scored by normalised regret against reference optima. On 60 random instances of an analytic family with a closed-form objective, ACRS attains a mean regret of 11.8% against 22.3-42.3% for Bayesian optimisation, cross-entropy, kernel-density, stochastic-gradient and refinement-only baselines; its lead is largest at small budgets and, relative to the kernel-density method, tends to grow with the dimension. ACRS also has the lowest mean regret on two six-dimensional benchmarks and three development problems, and performs comparably to pure local refinement on three held-out problems frozen before testing.

The primary analysis for longitudinal randomized controlled trials (RCTs) often compares treatment groups at the last timepoint, referred to as the landmark time. Assuming data are normally distributed and missing at random, the mixed model for repeated measures (MMRM) is widely used to conduct inference in terms of a mean difference. When outcomes violate normality assumption and/or the mean difference lacks a clear interpretation, we may quantify treatment effects using the probability that a treated participant would have a better outcome than (or win over) a control participant. For RCTs with missing data, one may apply the generalized pairwise comparison (GPC) procedure, which carries forward the results of a pairwise comparison from a previous timepoint. We propose first using ranks to converts each observation at a timepoint into a win fraction, reflecting the proportion of times that the observation is better than every observation in the comparison group. Then, we conduct inference for the win probability based on the win fractions using the MMRM to obtain the point and variance estimates. Simulation results suggest that our method performed much better than the GPC procedure. We illustrate our proposed procedure in SAS and R using data from two published trials.

We develop an iterative framework for Bayesian inference problems where the posterior distribution may involve computationally intensive models, intractable gradients, significant posterior concentration, and pronounced non-Gaussianity. Our approach integrates: (i) a generalized annealing scheme that combines geometric tempering with multi-fidelity modeling; (ii) expressive measure transport surrogates for the intermediate annealed and final target distributions, learned variationally without evaluating gradients of the target density; and, (iii) an importance-weighting scheme to combine multiple quadrature rules, which recycles and reweighs expensive model evaluations as successive posterior approximations are built. Our scheme produces both a quadrature rule for computing posterior expectations and a transport-based approximation of the posterior from which we can easily generate independent Monte Carlo samples. We demonstrate the efficiency and accuracy of our approach on low-dimensional but strongly non-Gaussian Bayesian inverse problems involving partial differential equations.

A common impediment in conducting inference for Bayesian nonparametric models is either the need for complex MCMC algorithms and/or computational run-time for large datasets. We propose solutions here for Enriched Dirichlet process mixtures (EDPM). We derive a variational Bayes estimator based on a previously developed truncation approximation for EDPMs. The variational Bayes estimator can be used in two ways: 1) to develop a more efficient truncation approximation; 2) as good initial values for a blocked Gibbs sampler based on this more efficient truncation approximation or for a polya urn sampler. We derive the accuracy of this more efficient truncation approximation and demonstrate how this allows for simple implementation of a blocked Gibbs Sampler EDPMs in Nimble. We confirm the validity of the approximations by simulations and illustrate on a real data set.

We study the aggregate hazard rate of a heterogeneous population whose individual event intensities are modeled as Cox (doubly stochastic) processes. In the deterministic hazard setting, the observed pool hazard is the survival weighted mean of the individual hazards, and its time derivative equals the mean individual hazard drift minus a variance term. This yields a transparent structural explanation of burnout in mortgage pools. We extend this perspective to stochastic intensity models. The observed pool hazard remains a survival-weighted mean, but now evolves as an Ito process whose drift contains the mean drift of the individual hazards and a negative selection term driven by cross-sectional dispersion, together with a diffusion term inherited from the common factor. We formulate the general identity and discuss special cases relevant to mortgage prepayment modeling.

Surface temperature is a fundamental Essential Climate Variable, serving as a primary indicator of climate change and exerting a profound influence on ecosystems, agriculture, and human livelihoods. Although existing research provides a foundation for understanding the climate of Ghana, there remains an opportunity to enhance this landscape with granular station-level analysis. Such high-resolution analysis complements existing studies by capturing localised climatic nuances. This study conducts a detailed spatio-temporal analysis of temperature trends across 22 meteorological stations from 1983 to 2021. Using daily maximum (Tmax) and minimum (Tmin) observations, data were subjected to quality control, homogeneity testing, and homogenisation according to World Meteorological Organisation (WMO) standards, using AgERA5 reanalysis as a reference.The significance and magnitude of trends were determined using the Modified Mann-Kendall test, which is robust in handling potential effects of autocorrelation, and Sen's slope estimator. Results revealed that temperature trends in Ghana are highly localised and seasonal, highlighting the necessity for more studies of this nature. A critical finding is the asymmetric warming across the country, with minimum temperatures rising at an accelerated rate compared to maximum temperatures. This narrowing of the diurnal temperature range poses significant threats to agricultural stability and public health because nocturnal cooling is diminished. These findings underscore the urgent need for site-specific, seasonal climate monitoring to inform customised adaptation strategies. To mitigate these impacts, the study recommends a robust policy framework focusing on afforestation and the transition to green energy.

History-dependent constitutive models serve as macroscopic closures for the aggregated effects of micromechanics. Their parameters are typically learned from experimental data. With a limited experimental budget, eliciting the full range of responses needed to characterize the constitutive relation can be difficult. As a result, the data can be well explained by a range of parameter choices, leading to parameter estimates that are uncertain or unreliable. To address this issue, we propose a Bayesian optimal experimental design framework to quantify, interpret, and maximize the utility of experimental designs for reliable learning of history-dependent constitutive models. In this framework, the design utility is defined as the expected reduction in parametric uncertainty or the expected information gain. This enables in silico design optimization using simulated data and reduces the cost of physical experiments for reliable parameter identification. We introduce two approximations that make this framework practical for advanced material testing with expensive forward models and high-dimensional data: (i) a Gaussian approximation of the expected information gain, and (ii) a surrogate approximation of the Fisher information matrix. The former enables efficient design optimization and interpretation, while the latter extends this approach to batched design optimization by amortizing the cost of repeated utility evaluations. Our numerical studies of uniaxial tests for viscoelastic solids show that optimized specimen geometries and loading paths yield image and force data that significantly improve parameter identifiability relative to random designs, especially for parameters associated with memory effects.

Diesel engine particulate matter (PM) is one of the most challenging emission constituents to predict. As engines become cleaner and emissions levels drop, manufacturers need reliable methods to quantify the PM generated by production engines. Due to the inaccuracy of commercial-grade sensors, they turn to predictive models to accurately estimate PM. In practice, this requires a computationally inexpensive model that provides PM estimates with calibrated uncertainty. Complex, multiscale physics make mechanistic models intractable and traditional data-driven methods struggle in transient drive cycles due to the stochastic nature of PM generation. Leveraging recent innovations in PM measurement technology, we introduce a novel PM model based on the Ornstein-Uhlenbeck (OU) process. The OU process is a mean-reverting stochastic process commonly used in financial modeling, now being explored for engineering applications, and can be described as a stochastic differential equation (SDE). We modify the OU process by parameterizing the terms of the SDE as functions of the engine state, which are then fit with a maximum likelihood estimate. In a synthetic example, we verify the ability of our model to learn a time-varying, parametrized OU process. We then train the model using real experimental data designed to dynamically cover the engine operating space and test the trained model on EPA-regulated drive cycles. For most drive cycles, we find the method accurately predicts cumulative output of PM across time.

Understanding covariate-varying interdependencies among features is of great interest in various applications. Motivated by microbiome studies where microbial abundances and interactions vary with environmental factors, we develop a Bayesian covariate-varying factor model. This model flexibly estimates heteroscedasticity in the covariance matrix as a function of covariates. Specifically, our approach employs covariance regression through linear regression on a lower-dimensional factor loading matrix. This formulation, combined with joint sparsity induced by the Dirichlet–Horseshoe prior for the factor loadings, provides robust estimation of covariate-varying covariance in high-dimensional settings. The model simultaneously incorporates a regression structure for the mean abundance and jointly addresses the covariate-varying mean and covariance structure. Furthermore, the model tackles key statistical challenges such as discreteness, over-dispersion, compositionality, and high dimensionality, common in microbiome data analysis, using a flexible nonparametric Bayesian framework. We thoroughly investigate the properties of the model and conduct extensive simulation studies to examine its performance. Real microbiome data examples are provided for illustration.

Collecting multiple types of data on the same set of subjects is common in modern scientific applications including, genomics, metabolomics, and neuroimaging. Joint and Individual Variance Explained (JIVE) seeks a low-rank approximation of the joint variation between two or more sets of features captured on common subjects and isolates this variation from that unique to eachset of features. We develop an expectation-maximization (EM) algorithm to estimate a probabilistic model for the JIVE framework. The model extends probabilistic principal components analysis to multiple data sets. Our maximum likelihood approach simultaneously estimates joint and individual components, which can lead to greater accuracy compared to other methods. We apply ProJIVE to measures of brain morphometry and cognition in Alzheimer's disease. ProJIVE learns biologically meaningful courses of variation, and the joint morphometry and cognition subject scores are strongly related to more expensive existing biomarkers. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Code to reproduce the analysis is available on our GitHub page.

Scientific discovery increasingly relies on AI systems to select candidates for expensive experimental validation, yet no principled, budget-aware evaluation framework exists for comparing selection strategies – a gap intensified by large language models (LLMs), which generate plausible scientific proposals without reliable downstream evaluation. We introduce the Budget-Sensitive Discovery Score (BSDS), a formally verified metric – 20 theorems machine-checked by the Lean 4 proof assistant – that jointly penalizes false discoveries (lambda-weighted FDR) and excessive abstention (gamma-weighted coverage gap) at each budget level. Its budget-averaged form, the Discovery Quality Score (DQS), provides a single summary statistic that no proposer can inflate by performing well at a cherry-picked budget. As a case study, we apply BSDS/DQS to: do LLMs add marginal value to an existing ML pipeline for drug discovery candidate selection? We evaluate 39 proposers – 11 mechanistic variants, 14 zero-shot LLM configurations, and 14 few-shot LLM configurations – using SMILES representations on MoleculeNet HIV (41,127 compounds, 3.5% active, 1,000 bootstrap replicates) under both random and scaffold splits. Three findings emerge. First, the simple RF-based Greedy-ML proposer achieves the best DQS (-0.046), outperforming all MLP variants and LLM configurations. Second, no LLM surpasses the Greedy-ML baseline under zero-shot or few-shot evaluation on HIV or Tox21, establishing that LLMs provide no marginal value over an existing trained classifier. Third, the proposer hierarchy generalizes across five MoleculeNet benchmarks spanning 0.18%-46.2% prevalence, a non-drug AV safety domain, and a 9x7 grid of penalty parameters (tau >= 0.636, mean tau = 0.863). The framework applies to any setting where candidates are selected under budget constraints and asymmetric error costs.

This thesis presents Regenerative Rejection Sampling (RRS), a novel approximate sampling algorithm inspired by classical Rejection Sampling and Markov Chain Monte Carlo methods. The method constructs a continuous-time regenerative process whose stationary distribution coincides with a target density known only up to a normalizing constant. Unlike standard Rejection Sampling, RRS does not require the existence of a finite constant that upper-bounds the likelihood ratio. As a result, its total variation convergence rate remains exponential for a larger class of scenarios compared to, for example, the Independent Metropolis-Hastings sampler, which requires a finite bounding constant. To explain the workings of the method, we first present a detailed review of renewal and regenerative processes, including their limit theorems, stationary versions, and convergence properties under standard conditions. We explain a coupling proof for exponential convergence of regenerative processes, under the assumption of a spread-out cycle length distribution. We then introduce the RRS algorithm, and derive its convergence rate. Its performance is compared theoretically and empirically with classical MCMC methods. Numerical experiments demonstrate that RRS can exhibit lower autocorrelations and faster effective mixing, both in synthetic examples and in a Bayesian probit regression model applied to a real medical dataset. Moreover, if the algorithm is run until time t, we show that the usual order $O(1/t)$ results for the bias of the time-average estimators, is improved to a bias of $O(1/t^2)$ for the estimator constructed from the RRS method, and provide easy-to-estimate non-asymptotic bounds for this bias.

This paper introduces a comprehensive framework for complex-valued probability measures and explores their novel applications in information theory and statistical analysis. We define a complex probability measure as a phase-modulated extension of a classical probability measure. Building upon this foundation, we propose three fundamental information-theoretic quantities: complex entropy, which quantifies distribution uniformity through phase coherence; complex divergence, an asymmetric measure of dissimilarity between distributions; and the complex metric, a symmetric distance function satisfying the triangle inequality. We establish these concepts rigorously for both continuous and discrete probability distributions, proving key properties such as boundedness, continuity under total variation convergence, and clear extremal behaviors. A detailed comparative analysis with classical measures (Shannon entropy and Kullback-Leibler divergence) highlights the unique geometric and interpretive advantages of the proposed framework, particularly its sensitivity to distributional shape via a tunable phase parameter. We elucidate a profound formal analogy between the complex entropy integral and Feynman's path integral formulation of quantum mechanics, suggesting a deeper conceptual bridge. Finally, we demonstrate the practical utility of the complex metric through a detailed application in nonparametric two-sample hypothesis testing, outlining the testing procedure, advantages, limitations, and providing a conceptual simulation. This work opens new avenues for analyzing probability distributions through the lens of complex analysis and interference phenomena, with potential impacts across information theory, statistical inference, and machine learning.

Continual post-training of generative models is widely used, yet a principled understanding of when and why forgetting occurs remains limited. We develop theoretical results under a two-mode mixture abstraction (representing old and new tasks), proposed by Chen et al. (2025) (arXiv:2510.18874), and formalize forgetting in two forms: (i) mass forgetting, where the old mixture weight collapses to zero, and (ii) old-component drift, where an already-correct old component shifts during training. For equal-covariance Gaussian modes, we prove that forward-KL objectives trained on data from the new distribution drive the old weight to zero, while reverse-KL objectives converge to the true target (thereby avoiding mass forgetting) and perturb the old mean only through overlap-gated misassignment probabilities controlled by the Bhattacharyya coefficient, yielding drift that decays exponentially with mode separation and a locally well-conditioned geometry with exponential convergence. We further quantify how replay interacts with these objectives. For forward-KL, replay must modify the training distribution to change the population optimum; for reverse-KL, replay leaves the population objective unchanged but prevents finite-batch old-mode starvation through bounded importance weighting. Finally, we analyze three recently proposed near-on-policy post-training methods, SDFT (arxiv:2601.19897), TTT-Discover (arxiv:2601.16175), and OAPL (arxiv:2602.19362), via the same lens and derive explicit conditions under which each retains old mass and exhibits overlap-controlled drift. Overall, our results show that forgetting can by precisely quantified based on the interaction between divergence direction, geometric behavioral overlap, sampling regime, and the visibility of past behavior during training.

Bayesian optimal experimental design (BOED) provides a powerful, decision-theoretic framework for selecting experiments so as to maximise the expected utility of the data to be collected. In practice, however, its applicability can be limited by the difficulty of optimising the chosen utility. The expected information gain (EIG), for example, is often high-dimensional and strongly non-convex. This challenge is particularly acute in the batch setting, where multiple experiments are to be designed simultaneously. In this paper, we introduce a new approach to batch EIG-based BOED via a probabilistic lifting of the original optimisation problem to the space of probability measures. In particular, we propose to optimise an entropic regularisation of the expected utility over the space of design measures. Under mild conditions, we show that this objective admits a unique minimiser, which can be explicitly characterised in the form of a Gibbs distribution. The resulting design law can be used directly as a randomised batch-design policy, or as a computational relaxation from which a deterministic batch is extracted. To obtain scalable approximations when the batch size is large, we then consider two tractable restrictions of the full batch distribution: a mean-field family, and an i.i.d. product family. For the i.i.d. objective, and formally for its mean-field extension, we derive the corresponding Wasserstein gradient flow, characterise its long-time behaviour, and obtain particle-based algorithms via space-time discretisations. We also introduce doubly stochastic variants that combine interacting particle updates with Monte Carlo estimators of the EIG gradient. Finally, we illustrate the performance of the proposed methods in several numerical experiments, demonstrating their ability to explore multimodal optimisation landscapes and obtain high-utility batches in challenging examples.

We mathematically prove that chemical reaction networks without hidden layers can solve tasks for which spiking neural networks require hidden layers. Our proof uses the deterministic mass-action kinetics formulation of chemical reaction networks. Specifically, we prove that a certain reaction network without hidden layers can learn a classification task previously proved to be achievable by a spiking neural network with hidden layers. We provide analytical regret bounds for the global behavior of the network and analyze its asymptotic behavior and Vapnik-Chervonenkis dimension. In a numerical experiment, we confirm the learning capacity of the proposed chemical reaction network for classifying handwritten digits in pixel images, and we show that it solves the task more accurately and efficiently than a spiking neural network with hidden layers. This provides a motivation for machine learning in chemical computers and a mathematical explanation for how biological cells might exhibit more efficient learning behavior within biochemical reaction networks than neuronal networks.

We study drift estimation for discretely observed high-dimensional Lévy-driven Ornstein–Uhlenbeck processes when the drift admits a low-rank-plus-sparse approximation. A localized, increment-truncated pseudo-likelihood is regularized by nuclear and entrywise $\ell_1$ norms. A weighted joint-cone argument and restricted cancellation condition yield non-asymptotic Frobenius oracle inequalities for exact and approximate structure. The stochastic complexity is $rd+s\log d$, with explicit discretization, truncation, and Lévy-regime sample-complexity terms. We derive operator- and infinity-norm score bounds from a Gaussian-width inequality and give a proximal-gradient algorithm. In 100-replicate experiments with mixed low-rank-plus-sparse drift, the estimator reduces mean relative Frobenius error by 4.4–5.9% versus localized Lasso and wins 89–92% of paired paths. Component diagnostics show that these gains concern recovery of the total drift rather than exact identification of its low-rank and sparse components.

Zero-shot text classification (ZSC) offers the promise of eliminating costly task-specific annotation by matching texts directly to human-readable label descriptions. While early approaches have predominantly relied on cross-encoder models fine-tuned for natural language inference (NLI), recent advances in text-embedding models, rerankers, and instruction-tuned large language models (LLMs) have challenged the dominance of NLI-based architectures. Yet, systematically comparing these diverse approaches remains difficult. Existing evaluations, such as MTEB, often incorporate labeled examples through supervised probes or fine-tuning, leaving genuine zero-shot capabilities underexplored. To address this, we introduce BTZSC, a comprehensive benchmark of 22 public datasets spanning sentiment, topic, intent, and emotion classification, capturing diverse domains, class cardinalities, and document lengths. Leveraging BTZSC, we conduct a systematic comparison across four major model families, NLI cross-encoders, embedding models, rerankers and instruction-tuned LLMs, encompassing 38 public and custom checkpoints. Our results show that: (i) modern rerankers, exemplified by Qwen3-Reranker-8B, set a new state-of-the-art with macro F1 = 0.72; (ii) strong embedding models such as GTE-large-en-v1.5 substantially close the accuracy gap while offering the best trade-off between accuracy and latency; (iii) instruction-tuned LLMs at 4–12B parameters achieve competitive performance (macro F1 up to 0.67), excelling particularly on topic classification but trailing specialized rerankers; (iv) NLI cross-encoders plateau even as backbone size increases; and (v) scaling primarily benefits rerankers and LLMs over embedding models. BTZSC and accompanying evaluation code are publicly released to support fair and reproducible progress in zero-shot text understanding.

We study trade-offs between the population risk curvature, geometry of the noise, and preconditioning on the generalisation ability of the multipass Preconditioned Stochastic Gradient Descent (PSGD). Many practical optimisation heuristics implicitly navigate this trade-off in different ways – for instance, some aim to whiten gradient noise, while others aim to align updates with expected loss curvature. When the geometry of the population risk curvature and the geometry of the gradient noise do not match, an aggressive choice that improves one aspect can amplify instability along the other, leading to suboptimal statistical behavior. In this paper we employ on-average algorithmic stability to connect generalisation of PSGD to the effective dimension that depends on these sources of curvature. While existing techniques for on-average stability of SGD are limited to a single pass, as first contribution we develop a new on-average stability analysis for multipass SGD that handles the correlations induced by data reuse. This allows us to derive excess risk bounds that depend on the effective dimension. In particular, we show that an improperly chosen preconditioner can yield suboptimal effective dimension dependence in both optimisation and generalisation. Finally, we complement our upper bounds with matching, instance-dependent lower bounds.