Papers

Although approaches to Independent Component Analysis (ICA) based on characteristic function seem theoretically elegant, they may suffer from implementational challenges because of numerical integration steps or selection of tuning parameters. Extending previously considered objective functions and leveraging results from the continuum Generalized Method of Moments of Carrasco and Florens (2000), I derive an optimal estimator that can take a tractable form and thus bypass these concerns. The method shares advantages with characteristic function approaches – it does not require the existence of higher-order moments or parametric restrictions – while retaining computational feasibility and asymptotic efficiency. The results are adapted to handle a possible first step that delivers estimated sensors. Finally, a by-product of the approach is a specification test that is valuable in many ICA applications. The method's effectiveness is illustrated through simulations, where the estimator outperforms efficient GMM, JADE, or FastICA, and an application to the estimation of Structural Vector Autoregressions (SVAR), a workhorse of the macroeconometric time series literature.

Theoretical guarantees are established for a standard estimator in a semi-parametric finite mixture model, where each component density is modeled as a product of univariate densities under a conditional independence assumption. The focus is on the estimator that maximizes a smoothed log-likelihood function, which can be efficiently computed using a majorization-minimization algorithm. This smoothed likelihood applies a nonlinear regularization operator defined as the exponential of a kernel convolution on the logarithm of each component density. Consistency of the estimators is demonstrated by leveraging classical M-estimation frameworks under mild regularity conditions. Subsequently, convergence rates for both finite- and infinite-dimensional parameters are derived by exploiting structural properties of the smoothed likelihood, the behavior of the iterative optimization algorithm, and a thorough study of the profile smoothed likelihood. This work provides the first rigorous theoretical guarantees for this estimation approach, bridging the gap between practical algorithms and statistical theory in semi-parametric mixture modeling.

One of the central challenges for instructors is offering meaningful individual feedback, especially in large courses. Faced with limited time and resources, educators are often forced to rely on generalized feedback, even when more personalized support would be pedagogically valuable. To overcome this limitation, one potential technical solution is to utilize large language models (LLMs). For an exploratory study using a new platform connected with LLMs, we conducted a LLM-corrected mock exam during the "Introduction to Statistics" lecture at the University of Munich (Germany). The online platform allows instructors to upload exercises along with the correct solutions. Students complete these exercises and receive overall feedback on their results, as well as individualized feedback generated by GPT-4 based on the correct answers provided by the lecturers. The resulting dataset comprised task-level information for all participating students, including individual responses and the corresponding LLM-generated feedback. Our systematic analysis revealed that approximately 7 % of the 2,389 feedback instances contained errors, ranging from minor technical inaccuracies to conceptually misleading explanations. Further, using a combined feedback framework approach, we found that the feedback predominantly focused on explaining why an answer was correct or incorrect, with fewer instances providing deeper conceptual insights, learning strategies or self-regulatory advice. These findings highlight both the potential and the limitations of deploying LLMs as scalable feedback tools in higher education, emphasizing the need for careful quality monitoring and prompt design to maximize their pedagogical value.

We present the first method for assessing the relevance of a model-based clustering result in a general framework. Standard validation criteria, like the adjusted Rand index, rely on external labels to assess partition accuracy; consequently, they are inapplicable to real-world clustering problems where labels are missing. In contrast, our method offers an internal goodness-of-fit diagnostic, since it evaluates the validity of the clustering mechanism by testing the specification of the posterior probabilities of classification defined on the unit simplex. Because this simplex dimension is fixed by the number of clusters, the procedure naturally circumvents the curse of dimensionality, making it applicable to high-dimensional data where traditional density-based tests fail. The testing procedure requires only a consistent estimator of the parameters and the associated posterior classification probabilities for each observation, and its implementation is straightforward, as no additional model fitting is needed. Under the null hypothesis, the method exploits the fact that any functional transformation of the posterior probabilities has the same expectation under both the model being tested and the true data-generating process. The resulting goodness-of-fit test is constructed via an empirical likelihood approach with a growing number of moment conditions, allowing asymptotic detection of any alternative. A block-splitting strategy, employed to account for parameter estimation, provides a vector of test statistics that behave like a vector of independent chi-square random variables. Therefore, the goodness-of-fit of the posterior classification probabilities is assessed via the goodness-of-fit of the vector of empirical likelihood ratio test statistics. Hence, based on the distribution of this vector of statistics, different goodness-of-fit tests (e.g., Kolmogorov-Smirnov) can be used to investigate the distribution of the vector of test statistics with an exact asymptotic significance level.

Deep ensembles are a powerful tool in machine learning, improving both model performance and uncertainty calibration. While ensembles are typically formed by training and tuning models individually, evidence suggests that jointly tuning the ensemble can lead to better performance. This paper investigates the impact of jointly tuning weight decay, temperature scaling, and early stopping on both predictive performance and uncertainty quantification. Additionally, we propose a partially overlapping holdout strategy as a practical compromise between enabling joint evaluation and maximizing the use of data for training. Our results demonstrate that jointly tuning the ensemble generally matches or improves performance, with significant variation in effect size across different tasks and metrics. We highlight the trade-offs between individual and joint optimization in deep ensemble training, with the overlapping holdout strategy offering an attractive practical solution. We believe our findings provide valuable insights and guidance for practitioners looking to optimize deep ensemble models. Code is available at: https://github.com/lauritsf/ensemble-optimality-gap

Replicability is central to scientific progress, and the partial conjunction (PC) hypothesis testing framework provides an objective tool to quantify it across disciplines. Existing PC methods assume independent studies. Yet many modern applications, such as genome-wide association studies (GWAS) with sample overlap, violate this assumption, leading to dependence among study-specific summary statistics. Failure to account for this dependence can drastically inflate type I errors when combining inferences. We propose e-Filter, a powerful procedure grounded on the theory of e-values. It involves a filtering step that retains a set of the most promising PC hypotheses, and a selection step where PC hypotheses from the filtering step are marked as discoveries whenever their e-values exceed a selection threshold. We establish the validity of e-Filter for FWER and FDR control under unknown study dependence. A comprehensive simulation study demonstrates its excellent power gains over competing methods. We apply e-Filter to a GWAS replicability study to identify consistent genetic signals for low-density lipoprotein cholesterol (LDL-C). Here, the participating studies exhibit varying levels of sample overlap, rendering existing methods unsuitable for combining inferences. A subsequent pathway enrichment analysis shows that e-Filter replicated signals achieve stronger statistical enrichment on biologically relevant LDL-C pathways than competing approaches.

The diffusion of ideas and language in society has conventionally been described by S-shaped models, such as the logistic curve. However, the role of sub-exponential growth – a slower-than-exponential pattern known in epidemiology – has been largely overlooked in broader social phenomena. Here, we present a piecewise power-law model to characterize complex growth curves with a few parameters. We systematically analyzed a large-scale dataset of approximately one billion Japanese blog articles linked to Wikipedia vocabulary, and observed consistent patterns in web search trend data (English, Spanish, and Japanese). Our analysis of 2,963 items, selected for reliable estimation (e.g., sufficient duration/peak, monotonic growth), reveals that 1,625 (55%) diffusion patterns without abrupt level shifts were adequately described by one or two segments. For single-segment curves, we found that (i) the mode of the shape parameter $\alpha$ was near 0.5, indicating prevalent sub-exponential growth; (ii) the peak diffusion scale is primarily determined by the growth rate $R$, with minor contributions from $\alpha$ or the duration $T$; and (iii) $\alpha$ showed a tendency to vary with the nature of the topic, being smaller for niche/local topics and larger for widely shared ones. Furthermore, a micro-behavioral model of outward (stranger) vs. inward (community) contact suggests that $\alpha$ can be interpreted as an index of the preference for outward-oriented communication. These findings suggest that sub-exponential growth is a common pattern of social diffusion, and our model provides a practical framework for consistently describing, comparing, and interpreting complex and diverse growth curves.

Transportability, the ability to maintain performance across populations, is a desirable property of markers of clinical outcomes. However, empirical findings indicate that markers often exhibit varying performances across populations. For prognostic markers that are advertised as predictive risk equations for an outcome of interest, oftentimes a form of updating is required when the equation is transported to populations with different outcome prevalences. Here, we revisit transportability of prognostic markers through the lens of the foundational framework of sufficient component causes (SCC). We argue that transporting a marker "as is" implicitly assumes predictive values are transportable, whereas conventional prevalence adjustment shifts the locus of transportability to accuracy metrics (sensitivity and specificity). Using a minimalist SCC framework that decomposes risk prediction into broad causal constituents, we show that both approaches rely on strong assumptions about the stability of cause distributions. An SCC framework instead invites making transparent assumptions about how different causes vary across populations, leading to different transportation methods. For example, in the absence of any external information other than outcome prevalence, an impartial perspective can assume all causes are responsible for change in prevalence, leading to a new form of marker transportation. Numerical experiments demonstrate that different transportability assumptions lead to varying degrees of information loss, depending on the distribution of causes across populations. An SCC perspective challenges common assumptions and practices for marker transportability, and results in novel transportability methods based on explicit assumptions on how different causes vary across populations.

This paper provides a graph-based characterization of partial regression coefficients in linear structural equation models. First, we derive a generalized graphical criterion that unifies the d-separation, single-door, and back-door criteria. This criterion provides a generically necessary and sufficient condition under which a partial regression coefficient coincides with a linear causal effect that is not mediated by other explanatory variables. Second, we clarify the mechanism underlying post-treatment bias and provide a quantitative characterization of this bias. This characterization offers a unified framework for analyzing graph structures that induce post-treatment bias, which have previously been studied on a case-by-case basis. These results are derived from the algebraic properties of acyclic directed mixed graphs and do not rely on any specific probability distribution. Consequently, they apply to a broad class of linear structural equation models.

In this paper, we study the problem of finding a collection of planted cycles in an \ER random graph $G \sim \mathcal{G}(n, \lambda/n)$, in analogy to the famous Planted Clique Problem. When the cycles are planted on a uniformly random subset of $\delta n$ vertices, we show that almost-exact recovery (that is, recovering all but a vanishing fraction of planted-cycle edges as $n \to \infty$) is information-theoretically possible if $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$ and impossible if $\lambda > \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. Moreover, despite the worst-case computational hardness of finding long cycles, we design a polynomial-time algorithm that attains almost exact recovery when $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. This stands in stark contrast to the Planted Clique Problem, where a significant computational-statistical gap is widely conjectured.

The proliferation of sixth-generation (6G) networks and the massive Internet of Things (IoT) demand wireless communication technologies that are ultra-low-power, secure, and covert. Noise-based communication has emerged as a transformative paradigm that meets these demands by encoding information directly into the statistical properties of noise, rather than using traditional deterministic carriers. This survey provides a comprehensive synthesis of this field, systematically exploring its fundamental principles and key methodologies, including thermal noise modulation (TherMod), noise modulation (NoiseMod) and its variants, and the Kirchhoff-law-Johnson-noise (KLJN) secure key exchange. We address critical practical challenges such as channel estimation and hardware implementation, and highlight emerging applications in simultaneous wireless information and power transfer (SWIPT) and non-orthogonal multiple access (NOMA). Our analysis confirms that noise-based systems offer unparalleled advantages in energy efficiency and covertness, and we conclude by outlining future research directions to realize their potential for enabling the next generation of autonomous and secure wireless networks.

Decision trees are widely used in high-stakes fields like finance and healthcare due to their interpretability. This work introduces an efficient, scalable method for generating synthetic pre-training data to enable meta-learning of decision trees. Our approach samples near-optimal decision trees synthetically, creating large-scale, realistic datasets. Using the MetaTree transformer architecture, we demonstrate that this method achieves performance comparable to pre-training on real-world data or with computationally expensive optimal decision trees. This strategy significantly reduces computational costs, enhances data generation flexibility, and paves the way for scalable and efficient meta-learning of interpretable decision tree models.

An important question in deep learning is how higher-order optimization methods affect generalization. In this work, we analyze a stochastic Gauss-Newton (SGN) method with Levenberg-Marquardt damping and mini-batch sampling for training overparameterized deep neural networks with smooth activations in a regression setting. Our theoretical contributions are twofold. First, we establish finite-time convergence bounds via a variable-metric analysis in parameter space, with explicit dependencies on the batch size, network width and depth. Second, we derive non-asymptotic generalization bounds for SGN using uniform stability in the overparameterized regime, characterizing the impact of curvature, batch size, and overparameterization on generalization performance. Our theoretical results identify a favorable generalization regime for SGN in which a larger minimum eigenvalue of the Gauss-Newton matrix along the optimization path yields tighter stability bounds.

This paper introduces an approach to multi-stream quickest change detection and fault isolation for unnormalized and score-based statistical models. Traditional optimal algorithms in the quickest change detection literature require explicit pre-change and post-change distributions to calculate the likelihood ratio of the observations, which can be computationally expensive for higher-dimensional data and sometimes even infeasible for complex machine learning models. To address these challenges, we propose the min-SCUSUM method, a Hyvarinen score-based algorithm that computes the difference of score functions in place of log-likelihood ratios. We provide a delay and false alarm analysis of the proposed algorithm, showing that its asymptotic performance depends on the Fisher divergence between the pre- and post-change distributions. Furthermore, we establish an upper bound on the probability of fault misidentification in distinguishing the affected stream from the unaffected ones.

We introduce a density-power weighted variant of the Stein operator, called the $\gamma$-Stein operator, for robust inference with unnormalized probability models. The operator is motivated by the first variation of the $\gamma$-divergence under infinitesimal escort transport and weights the usual Stein field by a positive power of the model density. This weighting down-weights observations in low model-density regions, providing a principled robustness mechanism while retaining the normalizing-constant-free structure of score matching. We develop the resulting $\gamma$-score matching estimating equations and discuss their non-integrable, generalized-method-of-moments character. We further study two extensions: a $\gamma$-kernelized Stein discrepancy, interpreted as a robust diagnostic or contaminated-null goodness-of-fit procedure, and $\gamma$-Stein variational gradient descent for robust posterior approximation. Numerical examples on directional, mixture, and quartic-potential models illustrate the robustness–efficiency trade-off: positive $\gamma$ can stabilize inference under targeted contamination, whereas $\gamma=0$ remains preferable under clean well-specified models.

Inferring the infinitesimal rates of continuous-time Markov chains (CTMCs) is a central challenge in many scientific domains. This task is difficult because the number of rates grows quadratically with the state space, rates can be strongly dependent, and many transitions may be only partially observed. We introduce a Bayesian framework that models CTMC rates as flexible functions of covariates through Gaussian processes. This enables nonlinear covariate effects, improves inference by incorporating external information, and helps identify potential drivers of CTMC dynamics. For posterior inference, we use Hamiltonian Monte Carlo and develop scalable exact and approximate gradients for likelihoods involving repeated matrix exponentials. With $N$ observations and $K$ CTMC states, these gradients reduce the dominant cost of existing derivative calculations from $O(NK^3)$, with large constants, to $O(K^3+NK^2)$, with cheaper constants. We demonstrate the method in Bayesian phylogenetic and phylogeographic inference, where CTMCs are central, and show strong performance on synthetic and real datasets, including empirical quadratic scaling in $K$ even when $N<K$.

We address the problem of quickest change detection in Markov processes with unknown transition kernels. The key idea is to learn the conditional score $\nabla_{\mathbf{y}} \log p(\mathbf{y}|\mathbf{x})$ directly from sample pairs $( \mathbf{x},\mathbf{y})$, where both $\mathbf{x}$ and $\mathbf{y}$ are high-dimensional data generated by the same transition kernel. In this way, we avoid explicit likelihood evaluation and provide a practical way to learn the transition dynamics. Based on this estimation, we develop a score-based CUSUM procedure that uses conditional Hyvarinen score differences to detect changes in the kernel. To ensure bounded increments, we propose a truncated version of the statistic. With Hoeffding's inequality for uniformly ergodic Markov processes, we prove exponential lower bounds on the mean time to false alarm. We also prove asymptotic upper bounds on detection delay. These results give both theoretical guarantees and practical feasibility for score-based detection in high-dimensional Markov models.

We develop a high-dimensional scaling limit for Stochastic Gradient Descent with Polyak Momentum (SGD-M) and adaptive step-sizes. This provides a framework to rigourously compare online SGD with some of its popular variants. We show that the scaling limits of SGD-M coincide with those of online SGD after an appropriate time rescaling and a specific choice of step-size. However, if the step-size is kept the same between the two algorithms, SGD-M will amplify high-dimensional effects, potentially degrading performance relative to online SGD. We demonstrate our framework on two popular learning problems: Spiked Tensor PCA and Single Index Models. In both cases, we also examine online SGD with an adaptive step-size based on normalized gradients. In the high-dimensional regime, this algorithm yields multiple benefits: its dynamics admit fixed points closer to the population minimum and widens the range of admissible step-sizes for which the iterates converge to such solutions. These examples provide a rigorous account, aligning with empirical motivation, of how early preconditioners can stabilize and improve dynamics in settings where online SGD fails.

A unified framework is presented to study the two-sample Behrens–Fisher problem – testing equality of means when two normal populations have unequal, unknown variances – and a compact expression is derived for the null distribution of the classical test statistic. Our new approach involves a Mellin–Barnes factorization that decouples the square root of a weighted sum of independent chi-square variates, thereby collapsing a challenging two-dimensional integral to a tractable single-contour integral. Closing the contour yields a residue series that terminates whenever either sample's degrees of freedom is odd. A complementary Euler–Beta reduction identifies the density as a Gauss hypergeometric function with explicit parameters, yielding a numerically stable form that recovers Student's $t$ under equal variances. Ramanujan's master theorem supplies exact inverse-power tail coefficients, which bound Lugannani–Rice saddle-point approximation errors and support reliable tail analyses. The proposed framework reveals why hypergeometric structure appears, why certain finite-sum cases arise, and how one can pass from the bulk of the distribution to its tails without altering the analytic framework. Finally, it lets us tabulate exact two-sided critical values over a broad grid of sample sizes and variance ratios that reveal the parameter surface on which the well-known Welch's approximation switches from conservative to liberal, quantifying its maximum size distortion.

This article presents a pragmatic framework for making formal, utility-based decisions from statistical inferences. The method calculates an expected utility score for an intervention by combining Bayesian posterior probabilities of different effect magnitudes with points representing their practical value. A key innovation is a unified, non-arbitrary points scale (1-9 for small to extremely large) derived from a principle linking tangible outcomes across different effect types. This tangible scale enables a principled "trade-off" method for including values for loss aversion, side effects, and implementation cost. The framework produces a single, definitive expected utility score, and the initial decision is made by comparing the magnitude of this single score to a user-defined smallest important net benefit, a direct and intuitive comparison made possible by the scale's tangible nature. This expected utility decision is interpreted alongside clinical magnitude-based decision probabilities or credible interval coverage to assess evidence strength. Inclusion of a standard deviation representing individual responses to an intervention (or differences between settings with meta-analytic data) allows characterization of differences between individuals (or settings) in the utility score expressed as proportions expected to experience benefit, a negligible effect, and harm. These proportions provide context for the final decision about implementation. Users must perform sensitivity analyses to investigate the effects of systematic bias and of the subjective inputs on the final decision. This framework, implemented in an accessible spreadsheet, has not been empirically validated. It represents a tool in development, designed for practical decision-making from available statistical evidence and structured thinking about values of outcomes.

Federated Learning (FL) enables decentralized model training across multiple clients while optionally preserving data privacy. However, communication efficiency remains a critical bottleneck, particularly for large-scale models. In this work, we introduce two complementary algorithms: ProjFL, designed for unbiased compressors, and ProjFL+EF, tailored for biased compressors through an Error Feedback mechanism. Both methods rely on projecting local gradients onto a shared client-server subspace spanned by historical descent directions, enabling efficient information exchange with minimal communication overhead. We establish convergence guarantees for both algorithms under strongly convex, convex, and non-convex settings. Empirical evaluations on standard FL classification benchmarks with deep neural networks show that ProjFL and ProjFL+EF achieve accuracy comparable to existing baselines while substantially reducing communication costs.

Quantifying human flourishing, a multidimensional construct including happiness, health, purpose, virtue, relationships, and financial stability, is critical for understanding societal well-being beyond economic indicators. Existing measures often lack fine spatial and temporal resolution. Here we introduce the Human Flourishing Geographic Index (HFGI), derived from analyzing approximately 2.6 billion geolocated U.S. tweets (2013-2023) using fine-tuned large language models to classify expressions across 48 indicators aligned with Harvard's Global Flourishing Study framework plus attitudes towards migration and perception of corruption. The dataset offers monthly and yearly county- and state-level indicators of flourishing-related discourse, validated to confirm that the measures accurately represent the underlying constructs and show expected correlations with established indicators. This resource enables multidisciplinary analyses of well-being, inequality, and social change at unprecedented resolution, offering insights into the dynamics of human flourishing as reflected in social media discourse across the United States over the past decade.

We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(\cdot,\cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative Khintchine inequality to obtain upper and lower bounds depending only on scalar statistics of the kernel function and a “correlation kernel” matrix corresponding to $k(\cdot,\cdot)$. We then apply our method to provide new, tighter approximations for inner-product kernel matrices on general high-dimensional data, where the sample size and data dimension are polynomially related. Our method obtains simplified proofs of existing results that rely on the moment method and combinatorial arguments while also providing novel approximation results for the case of anisotropic Gaussian data. Finally, using similar techniques to our approximation result, we show a tighter lower bound on the bias of kernel regression with anisotropic Gaussian data.

Quantifying the frequency of compound flood depths is a fundamental challenge in low-gradient coastal watersheds, where flood hazards arise from the nonlinear interaction of storm surge, rainfall, and riverine flooding. Existing approaches often characterize either the joint occurrence of flood drivers or the flood response for prescribed events, but they do not derive the long-term frequency distribution of compound flood depths from probabilistic descriptions of rainfall and antecedent hydrologic conditions. Traditionally, coastal flood frequency has been quantified using the Joint Probability Method (JPM), which represents storm surge probabilistically. Although recent studies have incorporated rainfall into JPM-based analyses, rainfall is treated as a deterministic function of JPM storm characteristics rather than as a conditional probability distribution. Here, we extend the JPM by coupling its event-scale stochastic description of storm characteristics with probabilistic rainfall realizations and stochastic antecedent hydrologic conditions, thereby enabling propagation of these stochastic processes through the flood response to derive the compound flood-depth distribution. The framework provides a statistical basis for delineating compound flood transition zones, probabilistically attributing flood depths to hydrologic and coastal processes, and selecting response-based design storms for specified annual exceedance probabilities (or return periods). Application to the Lake Maurepas basin, Louisiana, shows that the statistically defined compound flood transition zone is more than twice the area identified from event-based analyses and that compound interactions increase flood depths by up to 0.7 m. This extended JPM establishes a probabilistic foundation for compound flood hazard assessment and response-based design.

Causal structure learning has long been the central task of inferring causal insights from data. Despite the abundance of real-world processes exhibiting higher-order mechanisms, however, an explicit treatment of interactions in causal discovery has received little attention. In this work, we focus on extending the causal additive model (CAM) to additive models with higher-order interactions. This second level of modularity we introduce to the structure learning problem is most easily represented by a directed acyclic hypergraph which extends the DAG. We introduce the necessary definitions and theoretical tools to handle the novel structure we introduce and then provide identifiability results for the hyper DAG, extending the typical Markov equivalence classes. We next provide insights into why learning the more complex hypergraph structure may actually lead to better empirical results. In particular, more restrictive assumptions like CAM correspond to easier-to-learn hyper DAGs and better finite sample complexity. We finally develop an extension of the greedy CAM algorithm which can handle the more complex hyper DAG search space and demonstrate its empirical usefulness in synthetic experiments.

We present a geometric framework for regression on structured high-dimensional data that shifts the analysis from the ambient space to a geometric object capturing the data's intrinsic structure. The method addresses a fundamental challenge in analyzing datasets with high ambient dimension but low intrinsic dimension, such as microbiome compositions, where traditional approaches fail to capture the underlying geometric structure. Starting from a k-nearest neighbor covering of the feature space, the geometry evolves iteratively through heat diffusion and response-coherence modulation, concentrating mass within regions where the response varies smoothly while creating diffusion barriers where the response changes rapidly. This iterative refinement produces conditional expectation estimates that respect both the intrinsic geometry of the feature space and the structure of the response.

We bring a control perspective to the problem of identifying paths of measures for sampling via dynamic measure transport (DMT). We highlight the fact that commonly used paths may be poor choices for DMT and connect existing methods for learning alternate paths to mean-field games. Based on these connections we pose a flexible family of optimization problems for identifying tilted paths of measures for DMT and advocate for the use of objective terms which encourage smoothness of the corresponding velocities. We present a numerical algorithm for solving these problems based on recent Gaussian process methods for solution of partial differential equations and demonstrate the ability of our method to recover more efficient and smooth transport models compared to those which use an untilted reference path.

We present a bifidelity Karhunen–Loève expansion (KLE) surrogate model for field-valued quantities of interest (QoIs) under uncertain inputs. The QoIs considered here are scalar fields. The approach combines the spectral efficiency of the KLE with polynomial chaos expansions (PCEs) to preserve an explicit mapping between input uncertainties and output fields. By coupling inexpensive low-fidelity (LF) simulations that capture dominant response trends with a limited number of high-fidelity (HF) simulations that correct for systematic bias, the proposed method can enable accurate and computationally affordable surrogate construction. To further improve surrogate accuracy, we develop an active learning strategy that adaptively selects new HF evaluations based on the surrogate's generalization error, estimated via cross-validation and modeled using Gaussian process regression. New HF samples are then acquired by maximizing an expected improvement criterion, targeting regions of high surrogate error. The resulting BF-KLE-AL framework is demonstrated on three examples of increasing complexity: a one-dimensional analytical benchmark, a two-dimensional convection-diffusion system, and a three-dimensional turbulent round jet simulation based on Reynolds-averaged Navier–Stokes (RANS) and enhanced delayed detached-eddy simulations (EDDES). The experiments show that bifidelity gains depend on LF accuracy, discrepancy approximation, and the allocation of simulation cost. Active learning improves prediction over random sampling in several settings, while the cost-matched comparisons identify both favorable regimes and cases where an HF-only surrogate is more accurate.

We study batched nonparametric contextual bandits under a margin condition when the margin parameter $\alpha$ is unknown. To capture the statistical cost of this ignorance, we introduce the regret inflation criterion, defined as the ratio between the regret of an adaptive algorithm and that of an oracle knowing $\alpha$. We show that the optimal regret inflation grows polynomially with the horizon $T$, with exponent given by the value of a convex optimization problem that depends on the dimension, smoothness, and number of batches $M$. Moreover, the minimizer of this optimization problem directly prescribes the batch allocation and exploration strategy of a rate-optimal algorithm. Building on this principle, we develop RoBIN (RObust batched algorithm with adaptive BINning), which achieves the optimal regret inflation up to polylogarithmic factors. These results reveal a new adaptivity barrier: under batching, adaptation to an unknown margin parameter inevitably incurs a polynomial penalty, sharply characterized by a variational problem. Remarkably, this barrier vanishes once the number of batches exceeds order $\log \log T$; with only a doubly logarithmic number of updates, one can recover the oracle regret rate up to polylogarithmic factors.

The Fréchet regression is a useful method for modeling random objects in a general metric space given Euclidean covariates. However, the conventional approach could be sensitive to outlying objects in the sense that the distance from the regression surface is large compared to the other objects. In this study, we develop a robust version of the global Fréchet regression by incorporating weight parameters into the objective function. We then introduce the Elastic net regularization, favoring a sparse vector of robust parameters to control the influence of outlying objects. We provide a computational algorithm to iteratively estimate the regression function and weight parameters, with providing a linear convergence property. We also propose the Bayesian information criterion to select the tuning parameters for regularization, which gives adaptive robustness along with observed data. The finite sample performance of the proposed method is demonstrated through numerical studies on matrix and distribution responses.