Papers

The performance of gradient-based optimization methods, such as standard gradient descent (GD), greatly depends on the choice of learning rate. However, it can require a non-trivial amount of user tuning effort to select an appropriate learning rate schedule. When such methods appear as inner loops of other algorithms, expecting the user to tune the learning rates may be impractical. To address this, we introduce AutoGD: a gradient descent method that automatically determines whether to increase or decrease the learning rate at a given iteration. We establish the convergence of AutoGD, and show that we can recover the optimal rate of GD (up to a constant) for a broad class of functions without knowledge of smoothness constants. Experiments on a variety of traditional problems and variational inference optimization tasks demonstrate strong performance of the method, along with its extensions to AutoBFGS and AutoLBFGS.

Benchmarking quantum devices is a foundational task for the sustained development of quantum technologies. However, accurate in situ characterization of large-scale quantum devices remains a formidable challenge: such systems experience many different sources of errors, and cannot be simulated on classical computers. Here, we introduce new benchmarking methods based on random circuit sampling (RCS), that substantially extend the scope of conventional approaches. Unlike existing benchmarks that report only a single quantity–the circuit fidelity–our framework extracts rich diagnostic information, including spatiotemporal error profiles, correlated and contextual errors, and biased readout errors, without requiring any modifications of the experiment. Furthermore, we develop techniques that achieve this task without classically intractable simulations of the quantum circuit, by leveraging side information, in the form of bitstring samples obtained from reference quantum devices. Our approach is based on advanced high-dimensional statistical modeling of RCS data. We sharply characterize the information-theoretic limits of error estimation, deriving matching upper and lower bounds on the sample complexity across all regimes of side information. We identify surprising phase transitions in learnability as the amount of side information varies. We demonstrate our methods using publicly available RCS data from a state-of-the-art superconducting processor, obtaining in situ characterizations that are qualitatively consistent yet quantitatively distinct from component-level calibrations. Our results establish both practical benchmarking protocols for current and future quantum computers and fundamental information-theoretic limits on how much can be learned from RCS data.

The rise of large-scale pretrained models has made it feasible to generate predictive or synthetic features at low cost, raising the question of how to incorporate such surrogate predictions into downstream decision-making. We study this problem in the setting of online linear contextual bandits, where contexts may be complex, nonstationary, and only partially observed. In addition to bandit data, we assume access to an auxiliary dataset containing fully observed contexts–common in practice since such data are collected without adaptive interventions. We propose PULSE-UCB, an algorithm that leverages pretrained models trained on the auxiliary data to impute missing features during online decision-making. We establish regret guarantees that decompose into a standard bandit term plus an additional component reflecting pretrained model quality. In the i.i.d. context case with Hölder-smooth missing features, PULSE-UCB achieves near-optimal performance, supported by matching lower bounds. Our results quantify how uncertainty in predicted contexts affects decision quality and how much historical data is needed to improve downstream learning.

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone–Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.

Modeling clinical time-series data is hampered by the challenge of capturing latent, time-varying dependencies among features. State-of-the-art approaches often rely on black-box mechanisms or simple aggregation, failing to explicitly model how the influence of one clinical variable propagates through others over time. We propose $\textbf{Chain-of-Influence (CoI)}$, an interpretable deep learning framework that constructs an explicit, time-unfolded graph of feature interactions. CoI enables the tracing of influence pathways, providing a granular audit trail that shows how any feature at any time contributes to the final prediction, both directly and through its influence on other variables. We evaluate CoI on mortality and disease progression tasks using the MIMIC-IV dataset and a chronic kidney disease cohort. Our framework achieves state-of-the-art predictive performance (AUROC of 0.960 on CKD progression and 0.950 on ICU mortality), with deletion-based sensitivity analyses confirming that CoI's learned attributions faithfully reflect its decision process. Through case studies, we demonstrate that CoI uncovers clinically meaningful, patient-specific patterns of disease progression, offering enhanced transparency into the temporal and cross-feature dependencies that inform clinical decision-making.

Seasonal forecast of Arctic sea ice concentration is key to mitigate the negative impact and assess potential opportunities posed by the rapid decline of sea ice coverage. Seasonal prediction systems based on climate models often show systematic biases and complex spatio-temporal errors that grow with the forecasts. Consequently, operational predictions are routinely bias corrected and calibrated using retrospective forecasts. For predictions of Arctic sea ice concentration, error corrections are mainly based on one-to-one post-processing methods including climatological mean or linear regression correction and, more recently, machine learning. Such deterministic adjustments are confined at best to the limited number of costly-to-run ensemble members of the raw forecast. However, decision-making requires proper quantification of uncertainty and likelihood of events, particularly of extremes. We introduce a probabilistic error correction framework based on a conditional Variational Autoencoder model to map the conditional distribution of observations given the biased model prediction. This method naturally allows for generating large ensembles of adjusted forecasts. We evaluate our model using deterministic and probabilistic metrics and show that the adjusted forecasts are better calibrated, closer to the observational distribution, and have smaller errors than climatological mean adjusted forecasts.

We investigate robust nonparametric regression in the presence of heavy-tailed noise, where the hypothesis class may contain unbounded functions and robustness is ensured via a robust loss function $\ell_\sigma$. Using Huber regression as a close-up example within Tikhonov-regularized risk minimization in reproducing kernel Hilbert spaces (RKHS), we address two central challenges: (i) the breakdown of standard concentration tools under weak moment assumptions, and (ii) the analytical difficulties introduced by unbounded hypothesis spaces. Our first message is conceptual: conventional generalization-error bounds for robust losses do not faithfully capture out-of-sample performance. We argue that learnability should instead be quantified through prediction error, namely the $L_2$-distance to the truth $f^\star$, which is $\sigma$-independent and directly reflects the target of robust estimation. To make this workable under unboundedness, we introduce a probabilistic effective hypothesis space that confines the estimator with high probability and enables a meaningful bias–variance decomposition under weak $(1+\epsilon)$-moment conditions. Technically, we establish new comparison theorems linking the excess robust risk to the $L_2$ prediction error up to a residual of order $\mathcal{O}(\sigma^{-2\epsilon})$, clarifying the robustness–bias trade-off induced by the scale parameter $\sigma$. Building on this, we derive explicit finite-sample error bounds and convergence rates for Huber regression in RKHS that hold without uniform boundedness and under heavy-tailed noise. Our study delivers principled tuning rules, extends beyond Huber to other robust losses, and highlights prediction error, not excess generalization risk, as the fundamental lens for analyzing robust learning.

Over the past couple of decades, many active learning acquisition functions have been proposed, leaving practitioners with an unclear choice of which to use. Bayesian active learning offers principled objectives with explainable intuition, including Expected Error Reduction (EER), Expected Predictive Information Gain (EPIG), and Bayesian Active Learning by Disagreements (BALD). A key challenge of such methods is the difficulty of extending to the batch setting, leading to either computational challenges (BatchBALD) or dramatic performance drops (top-$B$ selection). In this work, we extend active learning's Bayesian Decision Theory framework to derive Partial Batch Label Sampling (ParBaLS) for the EPIG algorithm. We show experimentally that ParBaLS EPIG gives superior performance for a variety of datasets with different budgets using Bayesian logistic regression on embeddings from large pre-trained models. Our code is available at https://github.com/ADDAPT-ML/ParBaLS.

Beginning in the 1970s, statistician-cum-logician Per Martin-Löf wrote a series of papers developing what became Martin-Löf type theory, realizing a system where the distinction between mathematics and programming disappears. Inspired by this vision, this paper introduces dependent type theory (of which Martin-Löf type theory is an example) to a statistical audience. Examples from statistics and probability theory demonstrate how dependent type theory and an algebraic perspective can unify the theoretical and computational concerns of statistics, ensuring rigorous, machine-checked proofs and executable software.

We address the problem of localizing the source of infection in an undirected, tree-structured network under a susceptible-infected outbreak model. The infection propagates with independent random time increments (i.e., edge-delays) between neighboring nodes, while only the infection times of a subset of nodes can be observed. We show that a reduced set of observers may be sufficient, in the statistical sense, to localize the source and characterize its identifiability via the joint Laplace transform of the observers' infection times. Using the explicit form of these transforms in terms of the edge-delay probability distributions, we propose scale-invariant least-squares estimators of the source. We evaluate their performance on synthetic trees and on a river network, demonstrating accurate localization under diverse edge-delay models. To conclude, we highlight overlooked technical challenges for observer-based source localization on networks with cycles, where standard spanning-tree reductions may be ill-posed.

To define a steepest descent method over a neural network, we need to choose a norm for each layer, a way to aggregate these norms across layers, and whether to use normalization. We systematically explore different alternatives for aggregating norms across layers, both formalizing existing combinations of Adam and the recently proposed Muon as a type of non-Euclidean gradient descent, and deriving new variants of the Muon optimizer. Through a comprehensive experimental evaluation of the optimizers within our framework, we find that Muon is sensitive to the choice of learning rate, whereas a new variant we call MuonMax is significantly more robust. We then show how to combine any non-Euclidean gradient method with model based momentum (known as Momo). The new Momo variants of Muon are significantly more robust to hyperparameter tuning, and often achieve a better validation score. Thus for new tasks, where the optimal hyperparameters are not known, we advocate for using Momo in combination with MuonMax to save on costly hyperparameter tuning.

Recent experiments in neuroscience reveal that task-relevant variables are often encoded in approximately orthogonal subspaces of neural population activity. These disentangled, or abstract, representations have been observed in multiple brain areas and across different species. These representations have been shown to support out of distribution generalization and rapid learning of novel tasks. The mechanisms by which these representations emerge remain poorly understood, especially in the case of supervised task behavior. Here, we show mathematically that abstract representations of latent variables are guaranteed to appear in the hidden layer of feedforward nonlinear networks when they are trained on tasks that depend directly on these latent variables. These learned abstract representations reflect the semantics of the input stimuli. To show this, we reformulate the usual optimization over the network weights into a mean field optimization problem over the distribution of neural preactivations. We then apply this framework to finite-width ReLU networks and show that the hidden layer of these networks will exhibit an abstract representation at all global minima of the task objective. Finally, we extend our findings to two broad families of activation functions as well as deep feedforward architectures. Together, our results provide an explanation for the widely observed abstract representations in both the brain and artificial neural networks. In addition, the general framework that we develop here provides a mathematically tractable toolkit for understanding the emergence of different kinds of representations in task-optimized, feature-learning network models.

The training of deep neural networks predominantly relies on a combination of gradient-based optimisation and back-propagation for the computation of the gradient. While incredibly successful, this approach faces challenges such as vanishing or exploding gradients, difficulties with non-smooth activations, and an inherently sequential structure that limits parallelisation. Lifted training methods offer an alternative by reformulating the nested optimisation problem into a higher-dimensional, constrained optimisation problem where the constraints are no longer enforced directly but penalised with penalty terms. This chapter introduces a unified framework that encapsulates various lifted training strategies, including the Method of Auxiliary Coordinates, Fenchel Lifted Networks, and Lifted Bregman Training, and demonstrates how diverse architectures, such as Multi-Layer Perceptrons, Residual Neural Networks, and Proximal Neural Networks fit within this structure. By leveraging tools from convex optimisation, particularly Bregman distances, the framework facilitates distributed optimisation, accommodates non-differentiable proximal activations, and can improve the conditioning of the training landscape. We discuss the implementation of these methods using block-coordinate descent strategies, including deterministic implementations enhanced by accelerated and adaptive optimisation techniques, as well as implicit stochastic gradient methods. Furthermore, we explore the application of this framework to inverse problems, detailing methodologies for both the training of specialised networks (e.g., unrolled architectures) and the stable inversion of pre-trained networks. Numerical results on standard imaging tasks validate the effectiveness and stability of the lifted Bregman approach compared to conventional training, particularly for architectures employing proximal activations.

Oversampling is one of the most widely used approaches for addressing imbalanced classification. The core idea is to generate additional minority samples to rebalance the dataset. Most existing methods, such as SMOTE, require converting categorical variables into numerical vectors, which often leads to information loss. Recently, large language model (LLM)-based methods have been introduced to overcome this limitation. However, current LLM-based approaches typically generate minority samples with limited diversity, reducing robustness and generalizability in downstream classification tasks. To address this gap, we propose a novel LLM-based oversampling method designed to enhance diversity. First, we introduce a sampling strategy that conditions synthetic sample generation on both minority labels and features. Second, we develop a new permutation strategy for fine-tuning pre-trained LLMs. Third, we fine-tune the LLM not only on minority samples but also on interpolated samples to further enrich variability. Extensive experiments on 10 tabular datasets demonstrate that our method significantly outperforms eight SOTA baselines. The generated synthetic samples are both realistic and diverse. Moreover, we provide theoretical analysis through an entropy-based perspective, proving that our method encourages diversity in the generated samples.

Time series forecasting (TSF) remains a challenging and largely unsolved problem in machine learning, despite significant recent efforts leveraging Large Language Models (LLMs), which predominantly rely on Transformer architectures. Empirical evidence consistently shows that even powerful Transformers often fail to outperform much simpler models, e.g., linear models, on TSF tasks; however, a rigorous theoretical understanding of this phenomenon remains limited. In this paper, we provide a theoretical analysis of Transformers' limitations for TSF through the lens of In-Context Learning (ICL) theory. Specifically, under AR($p$) data, we establish that: (1) Linear Self-Attention (LSA) models $\textit{cannot}$ achieve lower expected MSE than classical linear models for in-context forecasting; (2) as the context length approaches to infinity, LSA asymptotically recovers the optimal linear predictor; and (3) under Chain-of-Thought (CoT) style inference, predictions collapse to the mean exponentially. We empirically validate these findings through carefully designed experiments. Our theory not only sheds light on several previously underexplored phenomena but also offers practical insights for designing more effective forecasting architectures. We hope our work encourages the broader research community to revisit the fundamental theoretical limitations of TSF and to critically evaluate the direct application of increasingly sophisticated architectures without deeper scrutiny.

Fingerprinting radio frequency (RF) emitters typically involves finding unique characteristics that are featured in their received signal. These fingerprints are nuanced, but sufficiently detailed, motivating the pursuit of methods that can successfully extract them. The downstream task that requires the most meticulous RF fingerprinting (RFF) is known as specific emitter identification (SEI), which entails recognising each individual transmitter. RFF and SEI have a long history, with numerous defence and civilian applications such as signal intelligence, electronic surveillance, physical-layer authentication of wireless devices, to name a few. In recent years, data-driven RFF approaches have become popular due to their ability to automatically learn intricate fingerprints. They generally deliver superior performance when compared to traditional RFF techniques that are often labour-intensive, inflexible, and only applicable to a particular emitter type or transmission scheme. In this paper, we present a generic and versatile machine learning (ML) framework for data-driven RFF with several popular downstream tasks such as SEI, data association (EDA) and RF emitter clustering (RFEC). It is emitter-type agnostic. We then demonstrate the introduced framework for several tasks using real RF datasets for spaceborne surveillance, signal intelligence and countering drones applications.

Shaw and Stevens call for a new paradigm in climate science criticizes Large Scale Determinism in favor of (i) embracing discrepancies, (ii) embracing hierarchies, and (iii) create disruption while keeping interpretability. The last 20 years have seen a plethora of contributions relating complex networks with climate data and climate models. We provide a view of climate networks through a triad of frameworks and associated paradigms: (a) networks of data, where both (geographical) nodes and their links (arcs) are determined according to some metrics and/or statistical criteria; (b) climate data over networks, where the structure of the network (for both vertices and edges) is topologically pre-determined, and the climate variable is continuously defined over the (nonlinear) network; finally, (c) networks for data, referring to the huge machinery based on networks within the realm machine learning and statistics, with specific emphasis on their use for climate data. This paper is not a mere description of each element of the network triad, but rather a manifesto for the creation of three classes of fusions (we term them bridges). We advocate and carefully justify a fusion within to provide a corpus unicuum inside the network triad. We then prove that the fusion within is the starting point for a fusion between, where the network triad becomes a condition sine qua non for the implementation of the Shaw-Stevens agenda. We culminate with a meta fusion that allows for the creation of what we term a Shaw-Stevens network ecosystem.

Some applied researchers hesitate to use nonparametric methods, worrying that they will lose power in small samples or overfit the data when simpler models are sufficient. We argue that at least some of these concerns are unfounded when nonparametric models are strongly shrunk toward parametric submodels. We consider expanding a parametric model with a nonparametric component $r(x)$ that is heavily shrunk toward zero. This construction allows the model to adapt automatically: if the parametric model is correct, the nonparametric component disappears, recovering parametric efficiency, while if it is misspecified, the flexible component activates to capture the missing signal. We show that this adaptive behavior follows from simple and general conditions. Specifically, we prove that Bayesian nonparametric models anchored to linear regression, including variants of Gaussian process regression and Bayesian additive regression trees, consistently identify the correct parametric submodel when it holds and give asymptotically efficient inference for regression coefficients. In simulations, we find that the general BART model performs identically to correctly specified linear regression when the parametric model holds, and substantially outperforms it when nonlinear effects are present. This suggests a practical paradigm: defensive model expansion as a safeguard against model misspecification.

Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the variance of a sample scales as a power function of the mean of the sample. We study generalizations of Taylor's law in the context of heavy-tailed distributions with infinite mean and variance. We establish the probabilistic limit and analyze the associated convergence rates. Our results extend the existing literature by relaxing the i.i.d. assumption to accommodate dependence and heterogeneity among the random variables. This generalization enables application to dependent data such as time series and network-structured data. We support the theoretical developments by extensive simulations, and the practical relevance through applications to real network data.

We propose a generative multivariate posterior sampler via flow matching. It offers a simple training objective, and does not require access to likelihood evaluation. The method learns a dynamic, block-triangular velocity field in the joint space of data and parameters, which results in a deterministic transport map from a source distribution to the desired posterior. The inverse map, named vector rank, is accessible by reversibly integrating the velocity over time. It is advantageous to leverage the dynamic design: proper constraints on the velocity yield a monotone map, which leads to a conditional Brenier map, enabling a fast and simultaneous generation of Bayesian credible sets whose contours correspond to level sets of Monge-Kantorovich data depth. Our approach is computationally lighter compared to GAN-based and diffusion-based counterparts, and is capable of capturing complex posterior structures. Finally, frequentist theoretical guarantee on the consistency of the recovered posterior distribution, and of the corresponding Bayesian credible sets, is provided.

Real-world clinical problems are often characterized by multimodal data, usually associated with incomplete views and limited sample sizes in their cohorts, posing significant limitations for machine learning algorithms. In this work, we propose a Bayesian approach designed to efficiently handle these challenges while providing interpretable solutions. Our approach integrates (1) a generative formulation to capture cross-view relationships with a semi-supervised strategy, and (2) a discriminative task-oriented formulation to identify relevant information for specific downstream objectives. This dual generative-discriminative formulation offers both general understanding and task-specific insights; thus, it provides an automatic imputation of the missing views while enabling robust inference across different data sources. The potential of this approach becomes evident when applied to the multimodal clinical data, where our algorithm is able to capture and disentangle the complex interactions among biological, psychological, and sociodemographic modalities.

Set-based transformer models for amortized probabilistic inference and meta-learning, such as neural processes, prior-fitted networks, and tabular foundation models, excel at single-pass marginal prediction. However, many applications require joint distributions over multiple predictions. Purely autoregressive architectures generate these efficiently but sacrifice flexible set-conditioning. Obtaining joint distributions from set-based models requires re-encoding the entire context at each autoregressive step, which scales poorly. We introduce a causal autoregressive buffer that combines the strengths of both paradigms. The model encodes the context once and caches it; a lightweight causal buffer captures dependencies among generated targets, with each new prediction attending to both the cached context and all previously predicted targets added to the buffer. This enables efficient batched autoregressive sampling and joint predictive density evaluation. Training integrates set-based and autoregressive modes through masked attention at minimal overhead. Across synthetic functions, EEG time series, a Bayesian model comparison task, and tabular regression, our method closely matches the performance of full context re-encoding while delivering up to $20\times$ faster joint sampling and density evaluation, and up to $7\times$ lower memory usage.

Stochastic dynamical systems provide essential mathematical frameworks for modeling complex real-world phenomena. The Fokker-Planck-Kolmogorov (FPK) equation governs the evolution of probability density functions associated with stochastic system trajectories. Developing robust numerical methods for solving the FPK equation is critical for understanding and predicting stochastic behavior. Here, we introduce the distribution self-adaptive normalized physics-informed neural network (DSN-PINNs) for solving time-dependent FPK equations through the integration of soft normalization constraints with adaptive resampling strategies. Specifically, we employ a normalization-enhanced PINN model in a pretraining phase to establish the solution's global structure and scale, generating a reliable prior distribution. Subsequently, guided by this prior, we dynamically reallocate training points via weighted kernel density estimation, concentrating computational resources on regions most representative of the underlying probability distribution throughout the learning process. The key innovation lies in our method's ability to exploit the intrinsic structural properties of stochastic dynamics while maintaining computational accuracy and implementation simplicity. We demonstrate the framework's effectiveness through comprehensive numerical experiments and comparative analyses with existing methods, including validation on real-world economic datasets.

Parameter estimation and inference from complex survey samples typically focuses on global model parameters whose estimators have asymptotic properties, such as from fixed effects regression models. The central challenge is to both mitigate bias induced from potentially unbalanced samples and to incorporate adjustments for differences in effective sample size to get correct variance and interval estimates. We present a motivating example of Bayesian inference for a multi-level or mixed effects model in which estimates of both the local parameters (e.g. group level random effects) and the global parameters need to be adjusted for the complex sampling design. We evaluate the limitations of the survey-weighted pseudo-posterior and an existing automated post-processing method to improve the uncertainty quantification. We propose modifications to the automated process and demonstrate their improvements for multi-level models via a simulation study and a motivating example from the National Survey on Drug Use and Health. Reproduction examples are available from the authors and the updated R package is available via github:https://github.com/RyanHornby/csSampling

Engineering risk is concerned with the likelihood of failure and the scenarios when it occurs. The sensitivity of failure probability to change in system parameters is relevant to risk-informed decision making. Computing sensitivity is at least one level more difficult than the probability itself, which is already challenged by a large number of input random variables, rare events and implicit nonlinear `black-box' response. Finite difference with Monte Carlo probability estimates is spurious, requiring the number of samples to grow with the reciprocal of step size to suppress estimation variance. Many existing works gain efficiency by exploiting a specific class of input variables, sensitivity parameters, or response in its exact or surrogate form. For general systems, this work presents a theory and associated Monte Carlo strategy for computing sensitivity using response values and gradients with respect to sensitivity parameters. It is shown that the sensitivity at a given response threshold can be expressed via the expectation of response gradient conditional on the threshold. Determining the expectation requires conditioning on the threshold that is a zero-probability event, but it can be resolved by the concept of kernel smoothing. The proposed method offers sensitivity estimates for all response thresholds generated in a single Monte Carlo run. It is investigated in a number of examples featuring sensitivity parameters of different nature. As response gradient becomes increasingly available, it is hoped that this work can provide the basis for embedding sensitivity calculations with reliability in the same Monte Carlo run.

The vulnerability of machine learning models to adversarial attacks remains a critical societal security challenge. Traditional defenses, such as adversarial training, typically robustify models by minimizing a worst-case loss. These deterministic approaches do not account for uncertainty in the adversary's attack. While stochastic defenses placing a probability distribution on the adversary exist, they often lack statistical rigor and fail to make explicit their underlying assumptions. To resolve these issues, we introduce a formal Bayesian framework that models adversarial uncertainty through a stochastic channel, articulating all probabilistic assumptions. This yields two robustification strategies: a proactive defense enacted during training, aligned with adversarial training, and a reactive defense enacted during operations, aligned with adversarial purification. Several state-of-the-art defenses can be recovered as limiting cases of our model. We empirically validate our methodology, showcasing the benefits of explicitly modeling adversarial uncertainty.

Boxplots and related visualization methods are widely used exploratory tools for taking a first look at collections of univariate variables. In this note an extension is provided that is specifically designed to detect and display bimodality and multimodality when the data warrant it. For this purpose a univariate clustering method is constructed that ensures contiguous clusters, meaning that no cluster has members inside another cluster, and such that each cluster contains at least a given number of unique members. The resulting bixplot display facilitates the identification and interpretation of potentially meaningful subgroups underlying the data. The bixplot also displays the individual data values, which can draw attention to isolated points. Implementations of the bixplot are available in both Python and R, and their many options are illustrated on several real datasets. For instance, an external variable can be visualized by color gradations inside the display.

Using i.i.d. data to estimate a high-dimensional distribution in Wasserstein distance is a fundamental instance of the curse of dimensionality. We explore how structural knowledge about the data-generating process which gives rise to the distribution can be used to overcome this curse. More precisely, we work with the set of distributions of probabilistic graphical models for a given directed acyclic graph. It turns out that this knowledge is only helpful if it can be quantified, which we formalize via smoothness conditions on the transition kernels in the disintegration corresponding to the graph. In this case, we prove that the rate of estimation is governed by the local structure of the graph, more precisely by dimensions corresponding to single nodes together with their parent nodes. The precise rate depends on the exact notion of smoothness assumed for the kernels, where either weak (Wasserstein-Lipschitz) or strong (bidirectional Total-Variation-Lipschitz) conditions lead to different results. We prove sharpness under the strong condition and show that this condition covers, as a special case, distributions having a positive Lipschitz density.

The universal approximation property uniformly with respect to weakly compact families of measures is established for several classes of neural networks. To that end, we prove that these neural networks are dense in Orlicz spaces, thereby extending classical universal approximation theorems even beyond the traditional $L^p$-setting. The covered classes of neural networks include widely used architectures like feedforward neural networks with non-polynomial activation functions, deep narrow networks with ReLU activation functions and functional input neural networks.

Large reasoning models (LRMs) have achieved remarkable progress in complex problem-solving tasks. Despite this success, LRMs typically suffer from high computational costs during deployment, highlighting a need for efficient inference. A practical direction of efficiency improvement is to switch the LRM between thinking and non-thinking modes dynamically. However, such approaches often introduce additional reasoning errors and lack statistical guarantees for the performance loss, which are critical for high-stakes applications. In this work, we propose Probably Approximately Correct (PAC) reasoning that controls the performance loss under the user-specified tolerance. Specifically, we construct an upper confidence bound on the performance loss and determine a threshold for switching to the non-thinking model. Theoretically, using the threshold to switch between the thinking and non-thinking modes ensures bounded performance loss in a distribution-free manner. Our comprehensive experiments on reasoning benchmarks show that the proposed method can save computational budgets and control the user-specified performance loss.