Papers

This work is focused on constructing space-time covariance functions through a hierarchical mixture approach that can serve as building blocks for capturing complex dependency structures. This approach provides a unifying modeling framework that not only constructs a new class of asymmetric space-time covariance functions with closed-form expressions, but also provides corresponding space-time process representations, which further unify constructions for several existing space-time covariance models. This hierarchical mixture framework decomposes the complexity of model specification at different levels of hierarchy, for which parsimonious covariance models can be specified with simple mixing measures to yield flexible properties and closed forms. A characterization theorem is provided to study how the mixing measures determine the statistical properties of covariance functions. Several new covariance models resulting from this framework are discussed in terms of their practical usefulness. This hierarchical mixture approach is extended for nonstationary and multivariate settings. A theorem is also provided to construct valid asymmetric space-time covariance functions with arbitrary and possibly different degrees of smoothness and long-range dependence in space and in time. This generalized class also bridges a theoretical gap in using the Lagrangian reference framework. The superior performance of several new parsimonious covariance models over existing models is verified with the well-known Irish wind data and the U.S. air temperature data.

We study the problem of reconstructing an unknown probability distribution to high accuracy, given noisy information about its low-degree Fourier coefficients. Prior results for this basic problem have typically imposed strong modeling assumptions on the distribution, often requiring that it is a linear combination of spatially separated point sources. In this work, we analyze a very general version of the high-dimensional recovery problem, by considering completely general distributions over the $d$-dimensional torus $[0,1)^d$; we do not assume any spatial separation between point sources, or even that the distribution is a finite linear combination of point sources. The question naturally arises: what can be said about distribution recovery in such a general setting? - As a warm-up, we first give results for reconstructing distributions with respect to the Wasserstein distance. We establish essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $\kappa$ of the noise for which accurate reconstruction is possible. Roughly speaking, our results here show that for $d$-dimensional distributions, estimates of $\approx \exp(d)$ many Fourier coefficients are necessary and sufficient for accurate reconstruction under the Wasserstein distance. - As our main result, we define a new notion of "heavy hitter" reconstruction for distributions, which essentially amounts to achieving high-accuracy reconstruction of all "sufficiently dense" regions of the distribution. We give essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $\kappa$ of the noise for which accurate reconstruction is possible under this notion. Our results show that (in sharp contrast with Wasserstein reconstruction) accurate estimates of only $\approx \exp(\sqrt{d})$ many Fourier coefficients are necessary and sufficient for heavy hitter reconstruction.

The sim-to-real gap, where agents trained in a simulator face significant performance degradation during testing, is a fundamental challenge in reinforcement learning. Extansive works adopt the framework of distributionally robust RL, to learn a policy that acts robustly under worst case environment shift. Within this framework, our objective is to devise algorithms that are sample efficient with interactive data collection and large state spaces. By assuming d-rectangularity of environment dynamic shift, we identify a fundamental hardness result for learning in online Markov game, and address it by adopting minimum value assumption. Then, a novel least square value iteration type algorithm, DR-CCE-LSI, with exploration bonus devised specifically for multiple agents, is proposed to find an \episilon-approximate robust Coarse Correlated Equilibrium(CCE). To obtain sample efficient learning, we find that: when the feature mapping function satisfies certain properties, our algorithm, DR-CCE-LSI, is able to achieve \epsilon-approximate CCE with a regret bound of O{dHmin{H,1/min{\sigma_i}}\sqrt{K}}, where K is the number of interacting episodes, H is the horizon length, d is the feature dimension, and \simga_i represents the uncertainty level of player i. Our work introduces the first sample-efficient algorithm for this setting, matches the best result so far in single agent setting, and achieves minimax optimalsample complexity in terms of the feature dimension d. Meanwhile, we also conduct simulation study to validate the efficacy of our algorithm in learning a robust equilibrium.

Score-based generative models have recently attracted significant attention for their ability to generate high-fidelity data by learning maps from simple Gaussian priors to complex data distributions. A natural generalization of this idea to transformations between arbitrary probability distributions leads to the Schrödinger Bridge (SB) problem. However, SB solutions rarely admit closed-form expressios and are commonly obtained through iterative stochastic simulation procedures, which are computationally intensive and can be unstable. In this work, we introduce a unified closed-form framework for representing the stochastic dynamics of SB systems. Our formulation subsumes previously known analytical solutions including the Schrödinger Föllmer process and the Gaussian SB as specific instances. Notably, the classical Gaussian SB solution, previously derived using substantially more sophisticated tools such as Riemannian geometry and generator theory, follows directly from our formulation as an immediate corollary. Leveraging this framework, we develop a simulation-free algorithm that infers SB dynamics directly from samples of the source and target distributions. We demonstrate the versatility of our approach in two settings: (i) modeling developmental trajectories in single-cell genomics and (ii) solving image restoration tasks such as inpainting and deblurring. This work opens a new direction for efficient and scalable nonlinear diffusion modeling across scientific and machine learning applications.

Statistical inference in evolutionary models with site-dependence is a long-standing challenge in phylogenetics and computational biology. We consider the problem of approximating marginal sequence likelihoods under dependent-site models of biological sequence evolution. We prove an upper bound on the mixing time for a Markov chain Monte Carlo algorithm that samples the conditional distribution over latent sample paths, when the chain is initialized with a warm start. We then introduce a sequential Monte Carlo (SMC) algorithm for approximating the marginal likelihood, and show that our mixing time bound can be combined with recent importance sampling and finite-sample SMC results to obtain bounds on the finite sample approximation error of the resulting estimator. Our results show that the proposed SMC algorithm yields an efficient randomized approximation scheme for many practical problems of interest, and offers a significant improvement over a recently developed importance sampler for this problem. Our approach combines recent innovations in obtaining bounds for MCMC and SMC samplers, and may prove applicable to other problems of approximating marginal likelihoods and Bayes factors.

Bayesian optimal experimental design is a principled framework for conducting experiments that leverages Bayesian inference to quantify how much information one can expect to gain from selecting a certain design. However, accurate Bayesian inference relies on the assumption that one's statistical model of the data-generating process is correctly specified. If this assumption is violated, Bayesian methods can lead to poor inference and estimates of information gain. Generalised Bayesian (or Gibbs) inference is a more robust probabilistic inference framework that replaces the likelihood in the Bayesian update by a suitable loss function. In this work, we present Generalised Bayesian Optimal Experimental Design (GBOED), an extension of generalised Bayesian inference to the experimental design setting which achieves robustness in both design and inference. Using a generalisation of Bayesian decision theory, we derive a new acquisition function, the Gibbs expected information gain. Our empirical results demonstrate that GBOED can enhance robustness to outliers and incorrect assumptions about the outcome noise distribution.

Large language models (LLMs) have been proposed as scalable tools to address the gap between the importance of individualized written feedback and the practical challenges of providing it at scale. However, concerns persist regarding the accuracy, depth, and pedagogical value of their feedback responses. The present study investigates the extent to which LLMs can generate feedback that aligns with educational theory and compares techniques to improve their performance. Using mock in-class exam data from two consecutive years of an introductory statistics course at LMU Munich, we evaluated GPT-generated feedback against an established but expanded pedagogical framework. Four enhancement methods were compared in a highly standardized setting, making meaningful comparisons possible: Using a state-of-the-art model, zero-shot prompting, few-shot prompting, and supervised fine-tuning using Low-Rank Adaptation (LoRA). Results show that while all LLM setups reliably provided correctness judgments and explanations, their ability to deliver contextual feedback and suggestions on how students can monitor and regulate their own learning remained limited. Among the tested methods, zero-shot prompting achieved the strongest balance between quality and cost, while fine-tuning required substantially more resources without yielding clear advantages. For educators, this suggests that carefully designed prompts can substantially improve the usefulness of LLM feedback, making it a promising tool, particularly in large introductory courses where students would otherwise receive little or no written feedback.

We study confidence interval construction for linear regression under Huber's contamination model, where an unknown fraction of noise variables is arbitrarily corrupted. While robust point estimation in this setting is well understood, statistical inference remains challenging, especially because the contamination proportion is not identifiable from the data. We develop a new algorithm that constructs confidence intervals for individual regression coefficients without any prior knowledge of the contamination level. Our method is based on a Z-estimation framework using a smooth estimating function. The method directly quantifies the uncertainty of the estimating equation after a preprocessing step that decorrelates covariates associated with the nuisance parameters. We show that the resulting confidence interval has valid coverage uniformly over all contamination distributions and attains an optimal length of order $O(1/\sqrt{n(1-\epsilon)^2})$, matching the rate achievable when the contamination proportion $\epsilon$ is known. This result stands in sharp contrast to the adaptation cost of robust interval estimation observed in the simpler Gaussian location model.

We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit $\lambda$-dependence to quantify how much an algorithm's performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.

This paper develops a framework for incorporating prior information into sequential multiple testing procedures while maintaining asymptotic optimality. We define a weighted log-likelihood ratio (WLLR) as an additive modification of the standard LLR and use it to construct two new sequential tests: the Weighted Gap and Weighted Gap-Intersection procedures. We prove that both procedures provide strong control of the family-wise error rate. Our main theoretical contribution is to show that these weighted procedures are asymptotically optimal; their expected stopping times achieve the theoretical lower bound as the error probabilities vanish. This first-order optimality is shown to be robust, holding in high-dimensional regimes where the number of null hypotheses grows and in settings with random weights, provided that mild, interpretable conditions on the weight distribution are met.

Financial institutions deploy Large Language Models (LLMs) for reconciliations, regulatory reporting, and client communications, but nondeterministic outputs (output drift) undermine auditability and trust. We quantify drift across five model architectures (7B-120B parameters) on regulated financial tasks, revealing a stark inverse relationship: smaller models (Granite-3-8B, Qwen2.5-7B) achieve 100% output consistency at T=0.0, while GPT-OSS-120B exhibits only 12.5% consistency (95% CI: 3.5-36.0%) regardless of configuration (p<0.0001, Fisher's exact test). This finding challenges conventional assumptions that larger models are universally superior for production deployment. Our contributions include: (i) a finance-calibrated deterministic test harness combining greedy decoding (T=0.0), fixed seeds, and SEC 10-K structure-aware retrieval ordering; (ii) task-specific invariant checking for RAG, JSON, and SQL outputs using finance-calibrated materiality thresholds (plus or minus 5%) and SEC citation validation; (iii) a three-tier model classification system enabling risk-appropriate deployment decisions; and (iv) an audit-ready attestation system with dual-provider validation. We evaluated five models (Qwen2.5-7B via Ollama, Granite-3-8B via IBM watsonx.ai, Llama-3.3-70B, Mistral-Medium-2505, and GPT-OSS-120B) across three regulated financial tasks. Across 480 runs (n=16 per condition), structured tasks (SQL) remain stable even at T=0.2, while RAG tasks show drift (25-75%), revealing task-dependent sensitivity. Cross-provider validation confirms deterministic behavior transfers between local and cloud deployments. We map our framework to Financial Stability Board (FSB), Bank for International Settlements (BIS), and Commodity Futures Trading Commission (CFTC) requirements, demonstrating practical pathways for compliance-ready AI deployments.

We consider the maximization of $x^\top \theta$ over $(x,\theta) \in \mathcal{X} \times \Theta$, with $\mathcal{X} \subset \mathbb{R}^d$ convex and $\Theta \subset \mathbb{R}^d$ an ellipsoid. This problem is fundamental in linear bandits, as the learner must solve it at every time step using optimistic algorithms. We first show that for some sets $\mathcal{X}$ e.g. $\ell_p$ balls with $p>2$, no efficient algorithms exist unless $\mathcal{P} = \mathcal{NP}$. We then provide two novel algorithms solving this problem efficiently when $\mathcal{X}$ is a centered ellipsoid. Our findings provide the first known method to implement optimistic algorithms for linear bandits in high dimensions.

Diffusion models, though successful, are known to suffer from hallucinations that create incoherent or unrealistic samples. Recent works have attributed this to the phenomenon of mode interpolation and score smoothening, but they lack a method to prevent their generation during sampling. In this paper, we propose a post-hoc adjustment to the score function during inference that leverages the Laplacian (or sharpness) of the score to reduce mode interpolation hallucination in unconditional diffusion models across 1D, 2D, and high-dimensional image data. We derive an efficient Laplacian approximation for higher dimensions using a finite-difference variant of the Hutchinson trace estimator. We show that this correction significantly reduces the rate of hallucinated samples across toy 1D/2D distributions and a high-dimensional image dataset. Furthermore, our analysis explores the relationship between the Laplacian and uncertainty in the score.

We study the problem of learning policies that maximize cumulative reward while satisfying safety constraints, even when the real environment differs from a simulator or nominal model. We focus on robust constrained Markov decision processes (RCMDPs), where the agent must maximize reward while ensuring cumulative utility exceeds a threshold under the worst-case dynamics within an uncertainty set. While recent works have established finite-time iteration complexity guarantees for RCMDPs using policy optimization, their sample complexity guarantees remain largely unexplored. In this paper, we first show that Markovian policies may fail to be optimal even under rectangular uncertainty sets unlike the unconstrained robust MDP. To address this, we introduce an augmented state space that incorporates the remaining utility budget into the state representation. Building on this formulation, we propose a novel Robust constrained Value iteration (RCVI) algorithm with a sample complexity of $\mathcal{\tilde{O}}(|S||A|H^5/\epsilon^2)$ achieving at most $\epsilon$ violation using a generative model where $|S|$ and $|A|$ denote the sizes of the state and action spaces, respectively, and $H$ is the episode length. To the best of our knowledge, this is the first sample complexity guarantee for RCMDP. Empirical results further validate the effectiveness of our approach.

We study language generation in the limit, where an algorithm observes an adversarial enumeration of strings from an unknown target language $K$ and must eventually generate new, unseen strings from $K$. Kleinberg and Mullainathan [KM24] proved that generation is achievable in surprisingly general settings. But their generator suffers from “mode collapse,” producing from an ever-smaller subset of the target. To address this, Kleinberg and Wei [KW25] require the generator's output to be “dense” in the target language. They showed that generation with density, surprisingly, remains achievable at the same generality. Both results assume perfect data: no noisy insertions and no omissions. This raises a central question: how much contamination can generation tolerate? Recent works made partial progress on this question by studying (non-dense) generation with either finite amounts of noise (but no omissions) or omissions (but no noise). We characterize robustness under contaminated enumerations: 1. Generation under Contamination: Language generation in the limit is achievable for all countable collections iff the fraction of contaminated examples converges to zero. When this fails, we characterize which collections are generable. 2. Dense Generation under Contamination: Dense generation is strictly less robust to contamination than generation. As a byproduct, we resolve an open question of Raman and Raman [ICML25] by showing that generation is possible with only membership oracle access under finitely many contaminated examples. Finally, we introduce a beyond-worst-case model inspired by curriculum learning and prove that dense generation is achievable even with infinite contamination provided the fraction of contaminated examples converges to zero. This suggests curriculum learning may be crucial for learning from noisy web data.

The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and various results on the asymptotic achievement of the lower bound. More precisely, we show that Wasserstein projection estimators (and one-step approximations thereof) are asymptotically sensitivity-efficient under suitable assumptions, and that there exist Hodges-type estimators which exhibit super-sensitivity-efficiency. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.

In this paper we propose a sequential minimax optimization (SMO) method for solving a class of constrained bilevel optimization problems in which the lower-level part is a possibly nonsmooth convex optimization problem, while the upper-level part is a possibly nonconvex optimization problem. Specifically, SMO applies a first-order method to solve a sequence of minimax subproblems, which are obtained by employing a hybrid of modified augmented Lagrangian and penalty schemes on the bilevel optimization problems. Under suitable assumptions, we establish an operation complexity of $O(\varepsilon^{-7}\log\varepsilon^{-1})$ and $O(\varepsilon^{-6}\log\varepsilon^{-1})$, measured in terms of fundamental operations, for SMO in finding an $\varepsilon$-KKT solution of the bilevel optimization problems with merely convex and strongly convex lower-level objective functions, respectively. The latter result improves the previous best-known operation complexity by a factor of $\varepsilon^{-1}$. Preliminary numerical results demonstrate significantly superior computational performance compared to the recently developed first-order penalty method.

Bayesian inference requires determining the posterior distribution, a task that becomes particularly challenging when the dimension of the parameter space is large and unknown. This limitation arises in many physics problems, such as Mixture Models (MM) with an unknown number of components or the inference of overlapping signals in noisy data, as in the Laser Interferometer Space Antenna (LISA) Global Fit problem. Traditional approaches, such as product-space methods or Reversible-Jump Markov Chain Monte Carlo (RJMCMC), often face efficiency and convergence limitations. This paper presents samsara, a Continuous-Time Markov Chain Monte Carlo (CTMCMC) framework that models parameter evolution through Poisson-driven birth, death, and mutation processes. samsara is designed to sample models of unknown dimensionality. By requiring detailed balance through adaptive rate definitions, CTMCMC achieves automatic acceptance of trans-dimensional moves and high sampling efficiency. The code features waiting time weighted estimators, optimized memory storage, and a modular design for easy customization. We validate samsara on three benchmark problems: an analytic trans-dimensional distribution, joint inference of sine waves and Lorentzians in time series, and a Gaussian MM with an unknown number of components. In all cases, the code shows excellent agreement with analytical and Nested Sampling results. All these features push samsara as a powerful alternative to RJMCMC for large- and variable-dimensional Bayesian inference problems.

The ability to reason lies at the core of artificial intelligence (AI), and challenging problems usually call for deeper and longer reasoning to tackle. A crucial question about AI reasoning is whether models can extrapolate learned reasoning patterns to solve harder tasks with longer chain-of-thought (CoT). In this work, we present a theoretical analysis of transformers learning on synthetic state-tracking tasks with gradient descent. We mathematically prove how the algebraic structure of state-tracking problems governs the degree of extrapolation of the learned CoT. Specifically, our theory characterizes the length generalization of transformers through the mechanism of attention concentration, linking the retrieval robustness of the attention layer to the state-tracking task structure of long-context reasoning. Moreover, for transformers with limited reasoning length, we prove that a recursive self-training scheme can progressively extend the range of solvable problem lengths. To our knowledge, we provide the first optimization guarantee that constant-depth transformers provably learn $\mathsf{NC}^1$-complete problems with CoT, significantly going beyond prior art confined in $\mathsf{TC}^0$, unless the widely held conjecture $\mathsf{TC}^0 \neq \mathsf{NC}^1$ fails. Finally, we present a broad set of experiments supporting our theoretical results, confirming the length generalization behaviors and the mechanism of attention concentration.

Linear regression is frequently applied in a variety of domains, some of which might contain sensitive information. This necessitates that the application of these methods does not reveal private information. Differentially private (DP) linear regression methods, developed for this purpose, compute private estimates of the solution. These techniques typically involve computing a noisy version of the solution vector. Instead, we propose releasing private sketches of the datasets, which can then be used to compute an approximate solution to the regression problem. This is motivated by the sketch-and-solve paradigm, where the regression problem is solved on a smaller sketch of the dataset instead of on the original problem space. The solution obtained on the sketch can also be shown to have good approximation guarantees to the original problem. Various sketching methods have been developed for improving the computational efficiency of linear regression problems under this paradigm. We adopt this paradigm for the purpose of releasing private sketches of the data. We construct differentially private sketches for the problems of least squares regression, as well as least absolute deviations regression. We show that the privacy constraints lead to sketched versions of regularized regression. We compute the bounds on the regularization parameter required for guaranteeing privacy. The availability of these private sketches facilitates the application of commonly available solvers for regression, without the risk of privacy leakage.

Methicillin-resistant Staphylococcus aureus (MRSA) is a bacterium that leads to severe infections in hospitalized patients. Previous epidemiological research has focused on MRSA transmission, but few studies have examined the influence of both hospital-acquired MRSA (HA-MRSA) and community-acquired MRSA (CA-MRSA) on MRSA spread in hospitals. In this study, we present a unique compartmental model for studying MRSA transmission patterns in hospitals in Edmonton, Alberta. The model consists of susceptible individuals, patients who have been colonized or infected with HA-MRSA or CA-MRSA, and isolated patients. We first use Bayesian inference with Markov chain Monte Carlo (MCMC) algorithms to estimate the posterior mean of parameters in the full model using data from hospitals in Edmonton. Then we develop multiple sub-models with varying assumptions about the origin of new MRSA colonization. We also estimate transmission rates in hospitals.

Combining a continuous "slab" density with discrete "spike" mass at zero, spike-and-slab priors provide important tools for inducing sparsity and carrying out variable selection in Bayesian models. However, the presence of discrete mass makes posterior inference challenging. "Sticky" extensions to piecewise-deterministic Markov process samplers have shown promising performance, where sampling from the spike is achieved by the process sticking there for an exponentially distributed duration. As it turns out, the sampler remains valid when the exponential sticking time is replaced with its expectation. We justify this by mapping the spike to a continuous density over a latent universe, allowing the sampler to be reinterpreted as traversing this universe while being stuck in the original space. This perspective opens up an array of possibilities to carry out posterior computation under spike-and-slab type priors. Notably, it enables us to construct sticky samplers using other dynamics-based paradigms such as Hamiltonian Monte Carlo; in fact, original sticky process can be established as a partial position-momentum refreshment limit of our Hamiltonian sticky sampler. Our theoretical and empirical findings suggest these alternatives to be at least as efficient as the original sticky approach.

Overparameterized fully-connected neural networks have been shown to behave like kernel models when trained with gradient descent, assuming standard scaling conditions on the width, the learning rate, and the parameter initialization. In the limit of infinitely large widths and infinitesimal learning rate, the obtained kernel provides a description of the learned model's output via a closed-form solution dependent on the architecture and the activation function. The Neural Tangent Kernel, central to this description, remains constant throughout training, a phenomenon that is referred to as “lazy training” or within the “lazy regime”. Prior works show that the “lazy regime” leads to non-varying hidden neuron activations in infinitely-wide networks. Moreover, as infinitely-wide networks increase in depth, the Neural Tangent Kernel induces a closed-form solution that is data-independent, hence trivial. The Neural Tangent Kernel seemingly fails to describe the complexity of overparameterized neural networks on two distinct axes: large widths and large depths. In this work, we challenge these two conclusions and open the door to re-evaluating the Neural Tangent Kernel's role in describing the output of overparameterized neural networks. Specifically, we show experimentally that while deviations in the activations of individual hidden neurons vanish, the aggregate norm of these deviations does not. We support this finding with a theoretical result showing that the activations of the last hidden layer do not remain constant. Furthermore, we demonstrate that properly scaling the depth and stopping time in infinitely-wide ReLU networks yields a well-behaved, non-trivial output at large dataset sizes. We empirically evaluate the stability of this behavior on large datasets, and we describe the essential properties that enable the generalization of our results to other kernels.

In differential privacy, random noise is introduced to privatize summary statistics of a sensitive dataset before releasing them. The noise level determines the privacy loss, which quantifies how easily an adversary can detect a target individual's presence in the dataset using the published statistic. Most privacy analyses provide non-asymptotic upper bounds on the privacy loss which hold uniformly across all datasets. Sometimes, these bounds can be pessimistic on a given dataset. In such cases, it can be useful to complement these privacy bounds with sharp privacy characterizations that quantify a mechanism's exact privacy loss on a given dataset. With this goal, we study differentially private principal component analysis (PCA), where the goal is to privatize the leading principal components of a dataset with $n$ samples and $p$ features. We analyze the exponential mechanism and provide sharp asymptotic characterizations of its utility and privacy loss in the high-dimensional limit ($p \rightarrow \infty$). We show that in this limit, detecting a target individual's presence using privatized principal components is asymptotically equivalent to distinguishing between two Gaussians with different means, where the mean difference depends on certain spectral properties of the dataset. Our analysis combines the hypothesis-testing formulation of privacy guarantees proposed by Dong, Roth, and Su (2022) with Le Cam's contiguity arguments.

In this work, we systematically benchmark two recently developed deep density methods for nonlinear filtering. We model the filtering density of a discretely observed stochastic differential equation through the associated Fokker–Planck equation, coupled with Bayesian updates at discrete observation times. The two filters: the deep splitting filter and the deep backward stochastic differential equation filter, are both based on Feynman–Kac formulas, Euler–Maruyama discretizations and neural networks. The two methods are extended to logarithmic formulations providing sound, robust, and positivity-preserving density approximations in increasing state dimension. Comparing to the classical bootstrap particle filter and an ensemble Kalman filter, we benchmark the methods on numerous examples. In the low-dimensional examples the particle filters work well, but when we scale up to a partially observed $100$-dimensional Lorenz-96 model, the particle-based methods fail and the logarithmic deep backward stochastic differential equation filter prevails. In terms of computational efficiency, the deep density methods reduce inference time by roughly two to five orders of magnitude relative to the particle-based filters.

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the interpolation problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal interpolation of the functionals. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear interpolation of the functionals are proposed in the case where spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.

The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic process is considered. Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case spectral uncertainty, where spectral densities of the processes are not known exactly while some sets of admissible spectral densities of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral densities

This paper focuses on the problem of the mean square optimal estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of the sequence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible densities.

In many areas of systems biology, including virology, pharmacokinetics, and population biology, dynamical systems are commonly used to describe biological processes. These systems can be characterized by estimating their parameters from sampled data. The key problem is how to optimally select sampling points to achieve accurate parameter estimation. Classical approaches often rely on Fisher information matrix-based criteria such as A-, D-, and E-optimality, which require an initial parameter estimate and may yield suboptimal results when the estimate is inaccurate. This study proposes two simulation-based methods for optimal sampling design that do not depend on initial parameter estimates. The first method, E-optimal-ranking (EOR), employs the E-optimal criterion, while the second utilizes a Long Short-Term Memory (LSTM) neural network. Simulation studies based on the Lotka-Volterra and three-compartment models demonstrate that the proposed methods outperform both random selection and classical E-optimal design.

This paper introduces and analyzes a framework that accommodates general heterogeneity in regression modeling. It demonstrates that regression models with fixed or time-varying parameters can be estimated using the OLS and time-varying OLS methods, respectively, across a broad class of regressors and noise processes not covered by existing theory. The proposed setting facilitates the development of asymptotic theory and the estimation of robust standard errors. The robust confidence interval estimators accommodate substantial heterogeneity in both regressors and noise. The resulting robust standard error estimates coincide with White's (1980) heteroskedasticity-consistent estimator but are applicable to a broader range of conditions, including models with missing data. They are computationally simple and perform well in Monte Carlo simulations. Their robustness, generality, and ease of implementation make them highly suitable for empirical applications. Finally, the paper provides a brief empirical illustration.