Papers

Deep residual architectures, such as ResNet and the Transformer, have enabled models of unprecedented depth, yet a formal understanding of why depth is so effective remains an open question. A popular intuition, following Veit et al. (2016), is that these residual networks behave like ensembles of many shallower models. Our key finding is an explicit analytical formula that verifies this ensemble perspective, proving that increasing network depth is mathematically equivalent to expanding the size of this implicit ensemble. Furthermore, our expansion reveals a hierarchical ensemble structure in which the combinatorial growth of computation paths leads to an explosion in the output signal, explaining the historical necessity of normalization layers in training deep models. This insight offers a first principles explanation for the historical dependence on normalization layers and sheds new light on a family of successful normalization-free techniques like SkipInit and Fixup. However, while these previous approaches infer scaling factors through optimizer analysis or a heuristic analogy to Batch Normalization, our work offers the first explanation derived directly from the network's inherent functional structure. Specifically, our Residual Expansion Theorem reveals that scaling each residual module provides a principled solution to taming the combinatorial explosion inherent to these architectures. We further show that this scaling acts as a capacity controls that also implicitly regularizes the model's complexity.

Accurate estimation of treatment effects in online A/B testing is challenging with zero-inflated and skewed metrics. Traditional tests, like Welch's t-test, often lack sensitivity with heavy-tailed data due to their reliance on means, as opposed to e.g., percentiles. The Controlled Experiments Using Pre-experiment Data (CUPED) technique improves sensitivity by reducing variance, yet that variance reduction is insufficient for highly skewed metrics. Alternatively, Yuen's t-test uses trimmed means to robustly handle outliers and skewness. This paper introduces a method that combines the variance reduction of CUPED with the robustness of Yuen's t-test to enhance hypothesis testing sensitivity. Our novel approach integrates trimmed data in a principled manner, offering a framework that balances variance reduction with robust location measures. We demonstrate improved detection of significant effects with smaller sample sizes, enabling quicker experimental decisions without sacrificing statistical power. This work broadens the utility of controlled experiments in environments characterized by highly skewed or high-variance data.

The C statistic is a widely used likelihood-ratio statistic for model fitting and goodness-of-fit assessments with Poisson data in high-energy physics and astrophysics. Although it enjoys convenient asymptotic properties, the statistic is routinely applied in cases where its nominal null distribution relies on unwarranted assumptions. Because researchers do not typically carry out robustness checks, their scientific findings are left vulnerable to misleading significance calculations. With an emphasis on low-count scenarios, we present a comprehensive study of the theoretical properties of C statistics and related goodness-of-fit algorithms. We focus on common “plug-in” algorithms where moments of C are obtained by assuming the true parameter equals its estimate. To correct such methods, we provide a suite of new principled user-friendly algorithms and well-calibrated p-values that are ready for immediate deployment in the (astro)physics data-analysis pipeline. Using both theoretical and numerical results, we show (a) standard $\chi^2$-based goodness-of-fit assessments are invalid in low-count settings, (b) naive methods (e.g., vanilla bootstrap) result in biased null distributions, and (c) the corrected Z-test based on conditioning and high-order asymptotics gives the best precision with low computational cost. We illustrate our methods via a suite of simulations and applied astrophysical analyses. An open-source Python package is provided in a GitHub repository.

Regularization is a central tool for addressing ill-posedness in inverse problems and statistical estimation, with the choice of a suitable penalty often determining the reliability and interpretability of downstream solutions. While recent work has characterized optimal regularizers for well-specified data distributions, practical deployments are often complicated by distributional uncertainty and the need to enforce structural constraints such as convexity. In this paper, we introduce a framework for distributionally robust optimal regularization, which identifies regularizers that remain effective under perturbations of the data distribution. Our approach leverages convex duality to reformulate the underlying distributionally robust optimization problem, eliminating the inner maximization and yielding formulations that are amenable to numerical computation. We show how the resulting robust regularizers interpolate between memorization of the training distribution and uniform priors, providing insights into their behavior as robustness parameters vary. For example, we show how certain ambiguity sets, such as those based on the Wasserstein-1 distance, naturally induce regularity in the optimal regularizer by promoting regularizers with smaller Lipschitz constants. We further investigate the setting where regularizers are required to be convex, formulating a convex program for their computation and illustrating their stability with respect to distributional shifts. Taken together, our results provide both theoretical and computational foundations for designing regularizers that are reliable under model uncertainty and structurally constrained for robust deployment.

We introduce BLAST, Bayesian Linear regression with Adaptive Shrinkage for Transfer, a Bayesian multi-source transfer learning framework for high-dimensional linear regression. The proposed analytical framework leverages global-local shrinkage priors together with Bayesian source selection to balance information sharing and regularization. We show how Bayesian source selection allows for the extraction of the most useful data sources, while discounting biasing information that may lead to negative transfer. In this framework, both source selection and sparse regression are jointly accounted for in prediction and inference via Bayesian model averaging. The structure of our model admits efficient posterior simulation via a Metropolis-within-Gibbs sampling algorithm allowing full posterior inference for the target regression coefficients, making BLAST both computationally practical and inferentially straightforward. Our method achieves more accurate posterior inference for the target than regularization approaches based on target data alone, while offering competitive predictive performance and superior uncertainty quantification compared to current state-of-the-art transfer learning methods. We validate its effectiveness through extensive simulation studies and illustrate its analytical properties when applied to a case study on the estimation of tumor mutational burden from gene expression, using data from The Cancer Genome Atlas (TCGA).

Kernel change-point detection (KCPD) has become a widely used tool for identifying structural changes in complex data. While existing theory establishes consistency under independence assumptions, real-world sequential data such as text exhibits strong dependencies. We establish new guarantees for KCPD under $m$-dependent data: specifically, we prove consistency in the number of detected change points and weak consistency in their locations under mild additional assumptions. We perform an LLM-based simulation that generates synthetic $m$-dependent text to validate the asymptotics. To complement these results, we present the first comprehensive empirical study of KCPD for text segmentation with modern embeddings. Across diverse text datasets, KCPD with text embeddings outperforms baselines in standard text segmentation metrics. We demonstrate through a case study on Taylor Swift's tweets that KCPD not only provides strong theoretical and simulated reliability but also practical effectiveness for text segmentation tasks.

The COVID-19 pandemic presents challenges to both public health and the economy. Our objective is to examine how household expenditure, a significant component of private demand, reacts to changes in mobility. This investigation is crucial for developing policies that balance public health and the economic and social impacts. We utilize extensive scanner data from a major retail chain in India and Google mobility data to address this important question. However, there are a few challenges, including outcomes with excessive zeros and complicated correlations, time-varying confounding, and irregular observation times. We propose incorporating a multiplicative structural nested mean model with inverse intensity weighting techniques to tackle these challenges. Our framework allows semiparametric/nonparametric estimation for nuisance functions. The resulting rate doubly robust estimator enables the use of a conventional sandwich variance estimator without taking into account the variability introduced by these flexible estimation methods. We demonstrate the properties of our method theoretically and further validate it through simulation studies. Using the Indian consumer spending data and Google mobility data, our method reveals that the substantial reduction in mobility has a significant impact on consumers' fresh food expenditure.

Neural diffusion processes provide a scalable, non-Gaussian approach to modelling distributions over functions, but existing formulations are limited to single-task inference and do not capture dependencies across related tasks. We propose multi-task neural diffusion processes (MT-NDPs), an extension that incorporates a task encoder to enable task-conditioned probabilistic regression and few-shot adaptation across related functions. The task encoder extracts a low-dimensional representation from context observations and conditions the diffusion model on this representation, allowing information sharing across tasks while accommodating variable-sized context sets and preserving permutation invariance with respect to the ordering of context observations. We evaluate MT-NDPs through synthetic experiments and two real-world applications involving wind turbine data and school examination outcomes. The results demonstrate improved predictive accuracy relative to single-task diffusion baselines when context observations are available, alongside competitive performance against neural-process and Gaussian-process alternatives. In the wind-energy application, MT-NDPs achieve the lowest average coverage error among the evaluated models at larger context sizes, demonstrating improved predictive uncertainty calibration. Overall, our findings highlight the potential of task-conditioned diffusion models for uncertainty-aware multi-task regression and few-shot adaptation to previously unseen tasks.

The Weak form Estimation of Nonlinear Dynamics (WENDy) method is a recently proposed class of parameter estimation algorithms that exhibits notable noise robustness and computational efficiency. This work examines the coverage and bias properties of the original WENDy-IRLS algorithm's parameter and state estimators in the context of the following differential equations: Logistic, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark. The estimators' performance was studied in simulated data examples, under four different noise distributions (normal, log-normal, additive censored normal, and additive truncated normal), and a wide range of noise, reaching levels much higher than previously tested for this algorithm.

Open-source data offers a scalable and transparent foundation for estimating vehicle activity and emissions in urban regions. In this study, we propose a data-driven framework that integrates MOVES and open-source GPS trajectory data, OpenStreetMap (OSM) road networks, regional traffic datasets and satellite imagery-derived feature vectors to estimate the link level operating mode distribution and traffic emissions. A neural network model is trained to predict the distribution of MOVES-defined operating modes using only features derived from readily available data. The proposed methodology was applied using open-source data related to 45 municipalities in the Boston Metropolitan area. The "ground truth" operating mode distribution was established using OSM open-source GPS trajectories. Compared to the MOVES baseline, the proposed model reduces RMSE by over 50% for regional scale traffic emissions of key pollutants including CO, NOx, CO2, and PM2.5. This study demonstrates the feasibility of low-cost, replicable, and data-driven emissions estimation using fully open data sources.

Mixtures models are a cornerstone of Bayesian modelling, and it is well-known that sampling from the resulting posterior distribution can be a hard task. In particular, popular reversible Markov chain Monte Carlo schemes are often slow to converge when the number of observations $n$ is large. In this paper we introduce a novel and simple non-reversible sampling scheme for Bayesian mixture models (with fixed or varying number of components), which is shown to drastically outperform classical samplers in many scenarios of interest, especially during convergence phase and when components in the mixture have non-negligible overlap. At the theoretical level, we show that the performance of the proposed non-reversible scheme cannot be worse than the standard one, in terms of asymptotic variance, by more than a factor of four; and we provide a scaling limit analysis suggesting that the non-reversible sampler can reduce the convergence time from O$(n^2)$ to O$(n)$. We also discuss why the statistical features of mixture models make them an ideal case for the use of non-reversible discrete samplers.

LLM inference often generates a batch of candidates for a prompt and selects one via strategies like majority voting or Best-of- N (BoN). For difficult tasks, this single-shot selection often underperforms. Consequently, evaluations commonly report Pass@$k$: the agent may submit up to $k$ responses, and only the best of them is used when computing regret. Motivated by this, we study inference scaling in the more general Pass@$k$ inference setting, and prove that neither majority voting nor BoN exhibits the desirable scaling with $k$ and the sampling budget $N$. Combining the advantages of majority voting and BoN, we propose a new inference strategy called Best-of-Majority (BoM), with a pivotal step that restricts the candidates to the responses with high frequency in the $N$ samples before selecting the top-$k$ rewards. We prove that when the sampling budget is $N=\tilde\Omega(C^*)$, the regret of BoM is $O(\epsilon_{\mathrm{opt}}+\sqrt{\epsilon_{\mathrm{RM}}^2C^*/k})$, where $C^*$ is the coverage coefficient, $\epsilon_{\mathrm{RM}}$ is the estimation error of the reward model, and $\epsilon_{\mathrm{opt}}$ is the estimation error of reward at the optimal response. We further establish a matching lower bound, certifying that our algorithm is minimax optimal. Beyond optimality, BoM has a key advantage: unlike majority voting and BoN, its performance does not degrade when increasing $N$. Experimental results of inference on math problems show BoM outperforming both majority voting and BoN.

Safety evaluation is an essential component of clinical trials. To protect study participants, these studies often implement safety stopping rules that will halt the trial if an excessive number of toxicity events occur. Existing safety monitoring methods often treat these events as binary outcomes. A strategy that instead handles these as time-to-event endpoints can offer higher power and a reduced time to signal of excess risk, but must manage additional complexities including censoring and competing risks. We propose the TITE-Safety approach for safety monitoring, which incorporates time-to-event information while handling censored observations and competing risks appropriately. This strategy is applied to develop stopping rules using score tests, Bayesian beta-extended binomial models, and sequential probability ratio tests. The operating characteristics of these methods are studied via simulation for common phase 2 and 3 trial scenarios. Across simulation settings, the proposed techniques offer reductions in expected toxicities of 20% or more compared to binary data methods and maintain the type I error rate near the nominal level across various event time distributions. These methods are demonstrated through a redesign of the safety monitoring scheme for BMT CTN 0601, a single arm, phase 2 trial that evaluated bone marrow transplant as treatment for severe sickle cell disease. Our R package "stoppingrule" offers functions to construct and evaluate these stopping rules, providing valuable tools for trial design to investigators.

Prediction is a key issue in time series analysis. Just as classical mean regression models, classical autoregressive methods, yielding L$^2$ point-predictions, provide rather poor predictive summaries; a much more informative approach is based on quantile (auto)regression, where the whole distribution of future observations conditional on the past is consistently recovered. Since their introduction by Koenker and Xiao in 2006, autoregressive quantile autoregression methods have become a popular and successful alternative to the traditional L$^2$ ones. Due to the lack of a widely accepted concept of multivariate quantiles, however, quantile autoregression methods so far have been limited to univariate time series. Building upon recent measure-transportation-based concepts of multivariate quantiles, we develop here a nonparametric vector quantile autoregressive approach to the analysis and prediction of (nonlinear as well as linear) multivariate time series.

Learning rate warm-up – increasing the learning rate at the beginning of training – has become a ubiquitous heuristic in modern deep learning, yet its theoretical foundations remain poorly understood. In this work, we provide a principled explanation for why warm-up improves training. We rely on a generalization of the $(L_0, L_1)$-smoothness condition, which bounds local curvature as a linear function of the loss suboptimality and exhibits desirable closure properties. We show – both theoretically and empirically – that this condition is satisfied by common neural architectures and accurately captures the curvature of the optimization landscape early in training. Adapting the learning rate in response to this curvature condition naturally induces a warm-up-like schedule, and we show that this choice yields provably faster convergence guarantees than using a fixed learning rate. Experiments on language and vision models show that the resulting one-parameter warm-up schedule can match tuned linear warm-up and improve over no warm-up.

Modern regression analyses are often undermined by covariate measurement error, misspecification of the regression model, and misspecification of the measurement error distribution. We present, to the best of our knowledge, the first Bayesian nonparametric learning framework targeting total robustness to all three challenges in general nonlinear regression. Our framework places a joint Dirichlet process prior on the latent covariate–response distribution and updates it with posterior pseudo-samples of the latent covariates, so that inference is calibrated to the joint law. This yields estimators defined by minimizing the discrepancy between posterior realizations of the joint Dirichlet process and the model-implied joint distribution. We establish generalization bounds and provide a first proof of convergence and consistency of the resulting estimators under non-degenerate measurement error. A gradient-based implementation enables efficient computation; simulations and two real-data studies show improved stability to misspecification under increasing measurement error relative to recent Bayesian and frequentist alternatives.

We prove rates of convergence and robustness to prior misspecification within a Generalised Variational Inference (GVI) framework with bounded divergences. This addresses a significant open challenge for GVI and Federated GVI that employ a different divergence to the Kullback-Leibler under prior misspecification, operate within a subset of possible probability measures, and result in intractable posteriors. Our theoretical contributions extend to misspecified priors that lead to inconsistent Bayes posteriors. In particular, we are able to establish sufficient conditions for existence and uniqueness of GVI posteriors on arbitrary Polish spaces, prove that the GVI posterior measure concentrates on a neighbourhood of loss minimisers, and extend this to rates of convergence regardless of the prior measure.

Diffusion- and flow-based generative models have recently demonstrated strong performance in protein backbone generation tasks, offering unprecedented capabilities for de novo protein design. However, despite their generation quality, these models are constrained by slow sampling, often requiring hundreds of iterative steps. This computational bottleneck limits their practical utility in large-scale protein discovery, where thousands to millions of candidate structures are needed. To address this challenge, we explore the techniques of score distillation, which has shown great success in reducing the number of sampling steps in the vision domain while maintaining high generation quality. However, a straightforward adaptation of these methods results in unacceptably low designability. We introduce Score Protein identity Distillation (SPID), which resolves this incompatibility by combining few-step generation with inference-time noise scaling. SPID adapts the Score identity Distillation (SiD) framework to both diffusion- and flow-based models without requiring access to pretraining data. Applied to the Proteina flow-matching model, our 16-step generator achieves 94.4% designability, matching the 400-step teacher, while delivering more than a 20-fold reduction in effective backbone-generation time and maintaining comparable diversity and novelty. SPID generalizes across unconditional generation, fold-class conditional generation, and motif scaffolding, and extends to equivariant diffusion architectures, achieving significant reduction in generation time with comparable generation quality to the teacher in all tasks. The resulting reduction in inference cost could facilitate large-scale in silico protein design, thereby advancing diffusion-based models toward real-world protein engineering applications. The PyTorch implementation is available at https://github.com/LY-Xie/SiD_Protein

Upcoming astronomical surveys will produce petabytes of high-resolution images of the night sky, providing information about billions of stars and galaxies. Detecting and characterizing the astronomical objects in these images is a fundamental task in astronomy – and a challenging one, as most of these objects are faint and many visually overlap with other objects. We propose an amortized variational inference procedure to solve this instance of small-object detection. Our key innovation is a family of spatially autoregressive variational distributions that partition and order the latent space according to a $K$-color checkerboard pattern. By construction, the conditional independencies of this variational family mirror those of the posterior distribution. We fit the variational distribution, which is parameterized by a convolutional neural network, using neural posterior estimation (NPE) to minimize an expectation of the forward KL divergence. Using images from the Sloan Digital Sky Survey, our method achieves state-of-the-art performance. We further demonstrate that the proposed autoregressive structure greatly improves posterior calibration.

Spectral methods, also known as chaos expansions, are widely used in global sensitivity analysis (GSA), as they leverage orthogonal bases of L2 spaces to efficiently compute Sobol' indices, particularly in data-scarce settings. When derivatives of the model are available, a desirable property, both for modeling and GSA purposes, is for the derivatives of the basis functions to also form an orthogonal basis. We demonstrate that the only basis satisfying this property is the one associated with weighted Poincaré inequalities and Sturm–Liouville eigenvalue problems, which we call Poincaré basis. We also show that under certain conditions the Poincaré basis achieves the same convergence rate as the best polynomial approximation for classes of smooth functions. We then introduce a comprehensive framework for gradient-enhanced GSA that integrates recent advances both in the construction of the expansion - with gradient-enhanced regression - and in the construction of weights for derivative-based sensitivity analysis. Furthermore, the proposed methodology is applicable to a broad class of probability measures and various choices of weights. We illustrate its efficiency on a challenging flood modeling case study, where Sobol' indices are accurately estimated using limited data.

The Wasserstein barycenter is defined as the mean of a set of probability measures under the optimal transport metric, and has numerous applications spanning machine learning, statistics, and computer graphics. In practice these input measures are empirical distributions built from sensitive datasets, motivating a differentially private (DP) treatment. We present, to our knowledge, the first algorithms for computing Wasserstein barycenters under differential privacy. Empirically, on synthetic data, MNIST, and large-scale U.S. population datasets, our methods produce high-quality private barycenters with strong accuracy-privacy tradeoffs.

We propose new Markov chain Monte Carlo algorithms to sample a uniform distribution on a convex body $K$. Our algorithms are based on the proximal sampler, which uses Gibbs sampling on an augmented distribution and assumes access to the so-called restricted Gaussian oracle (RGO). The key contribution of this work is an efficient implementation of the RGO for uniform sampling on convex $K$ that goes beyond the membership-oracle model used in many classical and modern uniform samplers, and instead leverages richer oracle access commonly assumed in convex optimization. We implement the RGO via rejection sampling and access to either a projection oracle or a separation oracle on $K$. In both oracle models, we provide non-asymptotic complexity guarantees for obtaining unbiased samples, with accuracy quantified in Rényi divergence and $\chi^2$-divergence, and we support these theoretical guarantees with numerical experiments.

Hospitals face challenges in long-term intensive care unit (ICU) capacity planning under uncertain demand. Admission rates fluctuate over time, and LOS distributions vary with patient heterogeneity, hospital location, case mix, and clinical practice. Common approaches rely on steady-state queueing models or heuristic rules with fixed parameters, which often fail to capture real occupancy dynamics. The widely used 85% occupancy rule, for example, recommends keeping average utilization below this level to preserve responsiveness, yet it is grounded in stationary assumptions and may lack resilience in time-varying systems. Our analysis shows that even when long-run utilization targets are satisfied, daily occupancy often exceeds 100% capacity. We propose a data-driven framework to estimate ICU bed occupancy using an $M_t/G_t/\infty$ queueing model with time-varying arrival rates and empirically fitted LOS distributions. The approach combines statistical decomposition and parametric fitting to capture temporal patterns in admissions and LOS, and is applied to multi-year data from neonatal ICUs (NICUs) in Calgary. We evaluate capacity scenarios including average-based thresholds and Poisson-based surge estimates. Results show that static heuristics are inadequate under fluctuating demand and underscore the importance of modeling LOS variability when estimating bed needs. Although the case study focuses on NICUs, the framework has potential applicability to other ICU settings and provides interpretable, data-informed support for systems facing rising demand and constrained capacity.

Persistent demographic disparities have been identified in the treatment of patients seeking care in the emergency department (ED). These may be driven in part by subconscious biases, which providers themselves may struggle to identify. To better understand the operation of these biases, we performed a retrospective cross-sectional analysis using electronic health records describing 339,400 visits to the ED of a single US pediatric medical center between 2019-2024. Odds ratios were calculated using propensity-score matching. Analyses were adjusted for confounding variables, including chief complaint, insurance type, socio-economic deprivation, and patient comorbidities. We also trained a machine learning [ML] model on this dataset to identify predictors of admission. We found significant demographic disparities in admission (Non-Hispanic Black [NHB] relative to Non-Hispanic White [NHW]: OR 0.77, 95% CI 0.73-0.81; Hispanic relative to NHW: OR 0.80, 95% CI 0.76-0.83). We also identified disparities in individual decisions taken during the ED stay. For example, NHB patients were significantly less likely than NHW patients to be assigned an `emergent' triage acuity score of (OR 0.70, 95% CI 0.67-0.72), but emergent NHB patients were also significantly less likely to be admitted than NHW patients with the same triage acuity (OR 0.86, 95% CI 0.80-0.93). Demographic disparities were particularly acute wherever patients had normal vital signs, public insurance, moderate socio-economic deprivation, or a home address distant from the hospital. An ML model assigned higher importance to triage score for NHB than NHW patients when predicting admission, reflecting these disparities in assignment. We conclude that many visit characteristics, clinical and otherwise, may influence the operation of subconscious biases and affect ML-driven decision support tools.

We study the minimization of the non-convex and non-differentiable objective function $v \mapsto \mathrm{E} ( \| X - v \| \| X + v \| - \| X \|^2 )$ in $\mathbb{R}^p$. In particular, we show that its minimizers recover the first principal component direction of elliptically symmetric $X$ under specific conditions. The stringency of these conditions is studied in various scenarios, including a diverging number of variables $p$. We establish the consistency and asymptotic normality of the sample minimizer. We propose a Weiszfeld-type algorithm for optimizing the objective and show that it is guaranteed to converge in a finite number of steps. The results are illustrated with two simulations.

In this work, we explore how Neural Jump ODEs (NJODEs) can be used as generative models for Itô processes. Given (discrete observations of) samples of a fixed underlying Itô process, the NJODE framework can be used to approximate the drift and diffusion coefficients of the process. Under standard regularity assumptions on the Itô processes, we prove that, in the limit, we recover the true parameters with our approximation. Hence, using these learned coefficients to sample from the corresponding Itô process generates, in the limit, samples with the same law as the true underlying process. Compared to other generative machine learning models, our approach has the advantage that it does not need adversarial training and can be trained solely as a predictive model on the observed samples without the need to generate any samples during training to empirically approximate the distribution. Moreover, the NJODE framework naturally deals with irregularly sampled data with missing values as well as with path-dependent dynamics, allowing to apply this approach in real-world settings. In particular, in the case of path-dependent coefficients of the Itô processes, the NJODE learns their optimal approximation given the past observations and therefore allows generating new paths conditionally on discrete, irregular, and incomplete past observations in an optimal way.

Hyperparameters greatly impact models' capabilities; however, modern models are too large for extensive search. Instead, researchers design recipes that train well across scales based on their understanding of the hyperparameters. Despite this importance, few tools exist for understanding the hyperparameter loss surface. We discover novel structure in it and propose a new theory yielding such tools. The loss surface is complex, but as you approach the optimum simple structure emerges. It becomes characterized by a few basic features, like its effective dimension and the best possible loss. To uncover this asymptotic regime, we develop a novel technique based on random search. Within this regime, the best scores from random search take on a new distribution we discover. Its parameters are exactly the features defining the loss surface in the asymptotic regime. From these features, we derive a new asymptotic law for random search that can explain and extrapolate its convergence. These new tools enable new analyses, such as confidence intervals for the best possible performance or determining the effective number of hyperparameters. We make these tools available at https://github.com/nicholaslourie/opda .

Understanding how urban residents travel between neighborhoods and destinations is critical for transportation planning, mobility management, and public health. By mining historical origin-to-destination flow patterns with spatial, temporal, and functional relations among urban places, we estimate probabilities of visits from neighborhoods to specific destinations. These probabilities capture neighborhood-level contributions to citywide vehicular and foot traffic, supporting demand estimation, accessibility assessment, and multimodal planning. Particularly, we introduce VisitHGNN, a heterogeneous, relation-specific graph neural network designed to predict visit probabilities at individual Points of interest (POIs). POIs are characterized using numerical, JSON-derived, and textual attributes, augmented with fixed summaries of POI–POI spatial proximity, temporal co-activity, and brand affinity, while census block groups (CBGs) are described with 72 socio-demographic variables. CBGs are connected via spatial adjacency, and POIs and CBGs are linked through distance-annotated cross-type edges. Inference is constrained to a distance-based candidate set of plausible origin CBGs, and training minimizes a masked Kullback-Leibler (KL) divergence to yield probability distribution across the candidate set. Using weekly mobility data from Fulton County, Georgia, USA, VisitHGNN achieves strong predictive performance with mean KL divergence of 0.287, MAE of 0.008, Top-1 accuracy of 0.853, and R-square of 0.892, substantially outperforming pairwise MLP and distance-only baselines, and aligning closely with empirical visitation patterns (NDCG@50 = 0.966); Recall@5 = 0.611). The resulting distributions closely mirror observed travel behavior with high fidelity, highlighting the model's potential for decision support in urban planning, transportation policy, mobility system design, and public health.

O-RAN testing is becoming increasingly difficult with the exponentially growing number of performance measurements as the system grows more complex, with additional units, interfaces, applications, and possible implementations and configurations. To simplify the testing procedure and improve system design for O-RAN systems, it is important to identify the dependencies among various performance measurements, which are inherently time-series and can be modeled as realizations of random processes. While information theory can be utilized as a principled foundation for mapping these dependencies, the robust estimation of such measures for random processes from real-world data remains challenging. This paper introduces AMIF-MDS, which employs aggregate mutual Information in frequency (AMIF), a practical proxy for directed information (DI), to quantify similarity and visualize inter-series dependencies with multidimensional scaling (MDS). The proposed quantile-based AMIF estimator is applied to O-RAN time-series testing data to identify dependencies among various performance measures so that we can focus on a set of “core” performance measures. Applying density-based spatial clustering of applications with noise (DBSCAN) to the MDS embedding groups mutually informative metrics, organically reveals the link-adaptation indicators among other clusters, and yields a “core” performance measure set for future learning-driven O-RAN testing.

We consider statistical inference for a class of continuous semimartingale regression models based on high-frequency observations subject to contamination by finite-activity jumps and spike noise. By employing density-power weighting and Hölder-inequality-based normalization, we propose easy-to-implement, robustified versions of the conventional Gaussian quasi-maximum-likelihood estimator that require only a single tuning parameter. We prove their asymptotic mixed normality at the standard rate of $\sqrt{n}$. It is theoretically shown that these estimators are simultaneously robust against contamination in both the covariate and response processes. Additionally, under suitable conditions on the selection of the tuning parameter, the proposed estimators achieve the same asymptotic distribution as the conventional estimator in the contamination-free case. Illustrative simulation results highlight the estimators' insensitivity to the choice of the tuning parameter.