Nonparametric

Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.

This article presents with DYSANOS the first generative market model for smooth SANOS option surfaces for all strikes and expiries which are free of static arbitrage. Our model is designed to generate entire paths of daily spot and option prices for years in the future. We present a robust and useful if somewhat simplistic baseline hidden state generative model in the form of an AR(1) model. We discuss model setup, data pipeline, and training and investigate numerical resence of dynamic arbitrage. We illustrate model performance on Option Metrics' IvyDB S&P Index data from 2020 to~2025 and compare it to a pure implied-vol PCA model.

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon--Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert--Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

Minimax-optimal rates for multivariate distribution estimation are known to suffer from the curse of dimensionality. We propose a sparse Bayesian network approach in which each conditional probability is estimated using sparsity-aware conditional mean methods. The resulting estimator, \textit{BAyesian Network Distribution regression} (BAND), handles mixed data types in high-dimensional time series and achieves polynomial total variation convergence rates while allowing the feature dimension to grow polynomially with the sample size. These rates are substantially faster than the classical optimal rates for multivariate histogram density estimators that lack sparsity. Empirical evaluations show that BAND performs competitively for data sampling and confidence region forecasting against a range of state-of-the-art benchmarks.

Image inpainting aims to recover missing regions while preserving structural consistency. We propose a non-parametric method without network training based on data-guided stochastic dynamics. Starting from a masked image, the missing pixels are evolved through a reverse-time stochastic differential equation with a kernel-weighted correction estimated directly from a reference dataset. This empirical correction guides the reconstruction toward high-density regions of the data distribution without training a neural network or fitting a parametric density model. Experiments on MNIST, Fashion-MNIST, and MVTec show that the proposed method outperforms Mean Fill, Telea, and Navier-Stokes inpainting in PSNR, SSIM, and visual quality. On CelebA, it remains competitive and produces plausible completions for structure-sensitive occlusions. These results demonstrate the effectiveness of empirical reference statistics as a non-parametric prior for image inpainting.

Mixed continuous--categorical data pose a representation problem for continuous generative models. Flow Matching and Gaussian diffusion operate in Euclidean spaces, whereas categorical laws lie on probability simplices and may be highly imbalanced. We study a logit-coordinate framework that encodes categorical variables as smoothed natural parameters and combines them with transformed numerical variables. This yields common formulations of Logit Flow Matching and Logit Diffusion. We introduce a mixed-distribution discrepancy separating categorical marginal error from conditional continuous Wasserstein error, and derive stability bounds and imbalance-aware nonparametric rates linking vector-field or drift error to decoded mixed-distribution error. Controlled simulations show that scaled-logit coordinates improve or match one-hot coordinates, especially under severe rare-cell imbalance. Across four real-data benchmarks and ten splits per dataset, Logit FM improves the primary distributional metrics on three datasets and is comparable on Churn2; Block-Conditional Logit FM consistently improves the flat model; and Logit Diffusion generally improves over or matches One-Hot Diffusion.

B-spline regression constitutes a widely used framework for nonparametric modeling. The performance of this methodology depends on specifying the number and placement of changepoints, known as knots, prior to the estimation process. Such knot sequence determines the dimension of the B-spline basis used to represent the regression function and the number of coefficients to be estimated. Therefore, the knots' choice affects the model's flexibility, influencing its smoothness and goodness-of-fit. Traditionally, this problem has been addressed either by explicitly selecting knots, via knot-selection algorithms, or by regularization methods, such as P-splines, which automatically tune the regressor's smoothness. The latter have become the standard in generalized additive models (GAMs). In contrast, knot-selection techniques, frequently neglected because of computational or modeling limitations, provide certain advantages which can be valuable in some contexts. In this work, we introduce a novel explicit knot-selection technique for GAMs based on an extension of the adaptive splines (A-splines) knot selection methodology, combined with a customized Fellner-Schall scheme for tuning the associated parameters. Our approach is evaluated on various synthetic and real datasets and compared with P-splines and state-of-the-art knot-selection techniques. The results indicate comparable performance, while producing models built on a substantially smaller number of basis elements.

Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $φ$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.

Kernel regression with tensor trains and Hadamard overparameterization (KReTTaH) is introduced as a training-data-free, interpretable, and nonparametric framework for multi-way data imputation. The imputation problem is reformulated as regression in reproducing kernel Hilbert spaces (RKHS), where the tensor regression coefficients are explicitly constrained to lie on fixed-rank tensor-train (TT) manifolds and structured via Hadamard overparameterization to promote sparsity and high representational efficiency. Rather than relying on costly cross-validation, KReTTaH jointly optimizes the TT coefficient tensors and the kernel covariance matrices within a Riemannian product-manifold framework -- the former on fixed-rank TT manifolds, the latter on the manifold of positive-definite matrices -- thereby enabling automated kernel-hyperparameter selection. Numerical tests on two challenging applications -- imputation of high-dimensional functional magnetic resonance imaging (fMRI) data and recovery of missing edge flows in dynamic graphs -- demonstrate that KReTTaH consistently outperforms state-of-the-art tensor-, Bayesian-, and neural-network-based baselines in terms of modeling accuracy.

Circular data, representing angles or directions, are frequently encountered in computer vision, biology, geology, and meteorology. Traditional regression targets the conditional mean, which is often geometrically misleading for circular responses under multimodal, skewed, or asymmetric data structures. To address these limitations, a lightweight deep generative framework, namely ANGLE, is introduced for non-parametric distributional regression on the circle. The full conditional distribution of an angular response, given Euclidean and circular covariates, is learned through a generative map optimized via a generalized circular energy score (GCES) loss. Desirable theoretical properties, including the strict propriety of the loss and the rotational equivariance of the estimators, are established. Furthermore, both pre- and post-additive noise models are accommodated. A unified toolbox is provided for advancing previously underexplored challenges in circular statistics: extrapolation, sufficient dimension reduction, and conditional distribution equality testing. The framework's efficacy is demonstrated through extensive simulations and real-world applications. Specifically, the proposal is utilized for object pose estimation from imagery and wind direction prediction, which are integral to surveillance, autonomous vehicles, and energy systems, respectively. Superior predictive performance and robust uncertainty quantification of the proposed method in these tasks are revealed.

Randomized smoothing has emerged as a scalable technique for certifying the adversarial robustness of classifiers. However, its application to regression remains under-explored and faces unique challenges. Existing regression certificates rely on probabilistic acceptance regions and fail to exploit the local geometry of the function. In this work, we present a novel framework for certified robust regression that addresses these limitations. We derive a prediction-centered certificate that guarantees the stability of the smoothed model's prediction and ensures practical computability at test time. We investigate several alternatives for constructing these certificates by explicitly incorporating means, variances, and gradients. In particular, we demonstrate on the MNIST rotation task that utilizing gradient information yields significantly tighter robustness certificates compared to the current state-of-the-art, alpha-smoothing.

We ask whether interaction is necessary for order-optimal 1-bit mean estimation over nonparametric finite-moment classes. Adaptive threshold-query protocols achieve the order-optimal 1-bit minimax rate, and the same rate is attainable with general 1-bit queries using only one adaptive transition (i.e., two stages of querying). In the non-adaptive setting, threshold and interval queries are known to be highly suboptimal, but the case of arbitrary non-adaptive quantizers remains unresolved. Can such quantizers match the adaptive rate, yielding an optimal one-shot protocol? Or is the known two-stage estimator stage-optimal, with a single adaptive transition being necessary and sufficient?

Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.

Few-shot segmentation asks a model to delineate a target class in a query image from only a handful of annotated examples, a setting most acute in remote sensing, where labels are scarce and the imagery departs sharply from the natural images on which vision backbones are pretrained. Prevailing approaches either train a segmenter on labelled episodes, which raises accuracy within the training distribution but binds the model to it, or reduce each class to a lossy summary of frozen features, a single prototype, a few cluster prototypes, or a discrete clustering, none of which preserves the internal structure of a multimodal class. We argue that a class is better described by a distribution than by a point, and that frozen self-supervised features already carry enough structure to estimate that distribution directly. We introduce FROST, a training-free few-shot segmenter that treats the reference foreground and background as two point clouds on the unit sphere of frozen DINOv3 features and labels each query token by a nonparametric density ratio, with a threshold the Bayes rule fixes at zero under equal priors. Because the variance of a density estimate shrinks as its sample grows, the decision sharpens as references accumulate, and every remaining quantity from the kernel bandwidth to the spatial gate is read from the support set rather than tuned. We develop FROST for overhead imagery, where a class is typically a scatter of many small and dissimilar instances that a density tracks but a lossy summary blurs. Across seventeen remote-sensing benchmarks FROST surpasses both training-free and learning-based methods, leading by 5.6 mIoU from a single annotated example and widening its lead as the support set grows, all while remaining among the smallest models compared. Code is available at https://github.com/jhpark-ai/FROST.

We consider sparse multivariate stochastic systems that evolve in continuous time according to a causal mechanism and present methodology to recover the system's time-infinitesimal transition mechanism from mere cross-sectional data. This observational paradigm is motivated by applications such as gene expression analysis, where destructive experimental techniques may only allow recording data once over a cell's lifetime. Precisely, we assume the system follows a time-homogeneous diffusion process that has reached an equilibrium distribution at observation time. Further, we assume the causal mechanism is fully described by the diffusion drift, is acyclic, and its causal structure graph is known. In this setting, we prove that the full causal mechanism, i.e., the drift function, can be non-parametrically identified under a weak non-explosion criterion. We derive a non-parametric kernel estimator for this challenging inverse problem and prove its consistency. Moreover, we propose a cross-validation scheme for hyperparameter tuning, illustrate the behavior of our estimator in simulations, and we discuss connections with irreversible generative diffusion models and low-frequency sampled data.

As machine learning models and datasets continue to grow, developing complex models has become increasingly computationally demanding. Knowledge distillation reduces deployment cost by compressing a large, well-trained teacher model into a compact student model, but it does not address settings where constructing the teacher itself is the bottleneck. Motivated by this challenge, we introduce Knowledge Cascade (KCas), a reverse knowledge distillation framework that uses information from a small, inexpensive student model to guide the development of a more complex teacher model. Although this direction is counterintuitive because the teacher typically has greater representational capacity, we show that student-to-teacher transfer can be principled when supported by statistical scaling relationships. We first develop KCas for nonparametric multivariate functional estimation in reproducing kernel Hilbert spaces via smoothing splines, where selecting multiple smoothing parameters is a major computational bottleneck. KCas transfers student-selected smoothing parameters to the full-sample regime through asymptotic scaling laws, substantially reducing computational cost for high-dimensional and large-scale datasets while retaining theoretical guarantees. Beyond smoothing splines, we illustrate the same principle through kernel density estimation and deep learning hyperparameter transfer. Simulations and real-data experiments show that KCas achieves substantial computational savings while maintaining strong statistical performance, and can sometimes outperform the corresponding full-sample procedure.

In the era of big data, subsampling became a common practice in statistical learning. By selecting a subgroup of individuals based on which the learner is trained, subsampling aims at reducing the computational cost and time of the estimation step, and ideally leads to a decrease of its energy consumption and carbon footprint. This work focuses on a nonparametric setting, in which the hypotheses set lies in a reproducing kernel Hilbert space, and the estimator is a minimizer of an empirical risk reweighted à la Horvitz-Thompson. By studying the asymptotic properties of this estimator, we reveal an optimal subsampling scheme (regarding the trace of the covariance operator) and show that it can be used via plug-in. A numerical study on synthetic and real-world datasets shows the practicability and the benefit of the proposed approach.

Online learning in non-stationary streams is often formulated as tracking a point estimate, but many applications require predicting the full data-generating distribution. We study online distributional prediction under drift and adversarial corruption. Our approach represents each candidate law through a latent cluster geometry: a variable-size configuration of centers that organizes probability mass and induces a predictive distribution. A Gibbs quasi-posterior over these configurations yields an online predictor by posterior averaging, and the resulting variable-dimensional posterior can be sampled with reversible-jump MCMC. The method therefore avoids specifying a parametric streaming law while retaining a structured latent space for uncertainty, regularization, and comparison. We evaluate performance by cumulative Wasserstein-1 regret against the time-varying true law. The analysis separates two effects: corruption perturbs the loss-based posterior update, whereas drift makes long-horizon posterior memory stale. We address the latter with a restarted variant that temporally localizes the same quasi-Bayesian update. The resulting high-probability bounds decompose into a PAC-Bayesian complexity term, a corruption-sensitive posterior perturbation term, and a dynamic optimal-transport term driven by $A_T^{\mathrm{OT}}=\sum_{t=2}^T W_2^2(p_{t-1}^*,p_t^*)$. Under bounded support, stable latent geometry, predictive-map regularity, oracle realizability, localized restart windows, sublinear transport action, and sublinear corruption budget, the restarted predictor achieves sublinear cumulative Wasserstein regret. These guarantees require no parametric model for the stream, drift mechanism, or corruption process.

Detecting distributional differences between two independent samples is a fundamental problem in statistics and machine learning. Nonparametric two-sample testing provides a principled framework for determining whether two samples are drawn from the same underlying distribution, without assuming any specific parametric form for the distribution. In this study, we propose a new two-sample test statistic based on a newly introduced integral probability metric (IPM), using a specially designed parametric discriminator class with a single node of a neural network. We show that the resulting test statistic, called PReLU-IPM, is nonparametric and establish theoretical guarantees for the associated two-sample testing procedure, PReLU-TST, including its consistency and asymptotical equivalence to nonparametric IPM-based tests under regularity conditions. By analyzing multiple simulated and real benchmark datasets, we demonstrate that PReLU-TST achieves higher power across a range of alternatives or performs comparably to its competitors, for finite samples.

Although some very common test beds exist for assessing the performance of clustering methods, large scale benchmarking is typically limited to relatively simplistic simulation set-ups. Here we describe the production and curation of close to 3000 synthetic data sets, derived from more than 200 publicly available data sets; the majority of which arose from real-world applications. By fitting a flexible non-parametric distribution to each base data set we are able to retain much of the nuance in real-world data which is difficult to reproduce in standard simulations, while also producing data sets whose sizes are sometimes substantially greater than the data sets from which they are derived. The synthetic data sets, plus an accompanying R package, are available for download from https://github.com/DavidHofmeyr/ClusBench.

This paper presents a method of multivariate scattered data interpolation and approximation that produces optimal Lipschitz-continuous approximation, subject to the desired monotonicity constraints. This method relies on tight upper and lower approximations to the data, and is similar in its spirit to the nearest-neighbour approximation but does not suffer from discontinuities. Local Lipschitz interpolation and Lipschitz smoothing are also presented. This approach falls under the umbrella of instance-based approximation with no training phase, and it is suitable for GPU-based parallelisation. A Python GPU-friendly package LipFit which implements the methods discussed is discussed.

Tabular foundation models (TFMs) have made rapid progress in standard classification and regression, but time-to-event survival prediction tasks have remained largely untouched. Unlike in standard regression tasks, survival prediction models must account for censored data. Standard TFMs cannot handle natively censored data, leading to biased and inaccurate predictions, making them unsuitable for real-world applications. To overcome this fundamental limitation, we propose \texttt{SurvPFN}, a prior-data fitted network (PFN), for survival prediction tasks. We pretrain \texttt{SurvPFN} on millions of synthetic survival prediction tasks to learn survival via distributional regression that accounts for censored data. \texttt{SurvPFN} works by (1) generating data with Weibull event times and a non-informative censoring mechanism; (2) integrating a censored event indicator; and (3) minimizing a censored negative log-likelihood. On SurvSet, a collection of real-world survival tasks, \texttt{SurvPFN} is highly competitive with classical and deep survival baselines without per-dataset fitting, a survival-specific architecture, or feature engineering. We show that survival can be treated as a continuous-time distributional regression problem with censored loss, unlocking the power of PFNs for time-to-event predictions.

Survival Analysis (SA) is a statistical framework that models the time span until some event of interest occurs. Widely used in several domains, including healthcare and churn prediction, a central challenge in its applicability stems from the time of the event being partially observed or \emph{right-censoring}. Tabular Foundation Models (TFM) have attracted significant interest in recent years due to their ability to perform prediction tasks in a single forward pass, requiring no dataset-specific parameter fitting. Despite their success, their application to prediction tasks on time-to-event data remains difficult due to right censoring. In this work, we present a training-free method to survival regression by leveraging TFMs to both predict the time of the event and iteratively impute right-censored data. Our method uses a TFM to construct an Accelerated Failure Time (AFT) model requiring no training beyond fitting a single scalar parameter. Subsequently, by building on the Buckley-James estimator, we introduce a non-parametric in-context estimator for right-censored data. Our experiments on standard survival analysis benchmarks show that our method is competitive with several parametric and semi-parametric survival regression models that require training, including Cox regression and parametric AFT models.

Continuous-time models are a natural choice for irregular and asynchronous data. A central design choice is how to embed discrete observations into continuous time. Interpolation- and imputation-based embeddings reconstruct a continuous observation path, making the model sensitive to the choice of reconstruction. We show that this reconstruction step is unnecessary; under mild conditions, compact-set universality on the model input space transfers to the data space whenever the embedding from data to input is continuous and injective. Guided by this result, and building on the rectilinear control path for Neural Controlled Differential Equations (NCDEs), we introduce a continuous and injective embedding for Log-NCDEs, a universal class of continuous-time models. Our approach records observations as increments and composes them over arbitrary query intervals to directly form log-signatures. This provides interval-level summaries without first interpolating the observed variables, while supporting online computation. Experiments on synthetic controlled dynamics and real-world time-series datasets show that the representation is accurate, efficient, and robust to irregular, asynchronous, and sparse observations.

We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.

Machine learning systems increasingly face requirements to forget not only individual data points, but entire domains of information, such as toxic language, copyrighted corpora, or demographic biases. This raises a fundamental dilemma of statistical-computational tradeoffs: removing all samples from an unwanted domain may be computationally prohibitive, while randomly removing a subset may not provide distribution-level statistical guarantees. We propose a statistical framework for distributional unlearning, in which domains are modeled as probability distributions, and the goal is to remove a carefully chosen subset of samples that reduces the effect of an unwanted distribution while preserving performance on a desired one. We formalize this using a hypothesis test of the edited data with the desired and unwanted domains, leading to an interpretable and robust criterion for selecting samples to remove. Within this statistical framework, we characterize the fundamental region of the allowable edited data distributions and the removal-preservation Pareto frontier for a broad class of distribution families. This includes parametric families such as shifted Gaussians of arbitrary dimension, a one-dimensional location family with log-concave noise, and the one-dimensional Poisson family. It also includes nonparametric families such as the Gaussian white noise model, a canonical model for nonparametric regression. We prove composition rules that describe how distributional unlearning behaves across multimodal unwanted domains, and introduce a central-limit behavior for the removal-preservation baselines when composing a large number of such families. Finally, we provide finite sample guarantees by providing Pareto frontiers for some selection algorithms, and observe an information-computation gap.

We introduce an empowered transposed Fully Connected Weighted (t-FCW) graph representation to embed point clouds into a metric space. While original t-FCW has shown promising results for point cloud classification, the reasons behind its effectiveness and its broader applicability remained unclear. In this work, we analyze the properties that make the empowered and original t-FCW effective and design a network that uses the empowered t-FCW exclusively as feature extractors. From an interpretability perspective, we build memory banks for classification, part segmentation, and semantic segmentation using the empowered t-FCW. Our analysis reveals that the empowered t-FCW inherits robustness from surface descriptors, provides interpretability through dimension-wise relations. These properties enable a highly efficient and interpretable network, which processes the ModelNet40 classification problem in approximately 7 seconds on an NVIDIA RTX A5000 GPU. Importantly, empowered t-FCW can function both as a lightweight standalone baseline and as a complementary plug-in to existing deep models.

Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on Gaussian processes, kernel machines, linear models, or linearized neural approximations, leaving a gap between theory and the nonlinear models used in practice. We develop a kernel based framework for analyzing regularized nonlinear parametric models trained on adaptively collected data. Our approach uses kernels over the parameter space to induce reproducing kernel Hilbert space structures over the corresponding model class, yielding confidence bounds for models trained with broad classes of regularized convex losses. We show how these bounds can support convergence guarantees for nonlinear acquisition and surrogate models, including randomized regularized policies that select points by maximizing a trained random model. These results provide a unified route to analyzing nonlinear parametric models in Bayesian optimization and related adaptive optimization settings.

Reversible instance normalization (RevIN) and its successors (Dish-TS, SAN, FAN) have become the de facto plug-in for time-series forecasting, yet the map they apply to each data point is strictly affine, $x \mapsto ax+b$, so they cannot reshape the underlying distribution -- heavy tails remain heavy and skewness remains uncorrected. We propose NoRIN, a non-linear reversible normalization based on the arcsinh-form Johnson $S_U$ transform with two shape parameters $(δ,\varepsilon)$ that control tailedness and skewness; the linear $Z$-score used by RevIN is recovered only in the limit $δ\to \infty$. Training $(δ,\varepsilon)$ jointly with the backbone via gradient descent reliably pushes them toward this linear limit within a few epochs -- a phenomenon we name the degeneration problem: the forecasting loss is locally indifferent to shape, and the high-capacity backbone compensates for any monotone reparameterization of its input. NoRIN escapes the degeneration by decoupling shape selection from gradient training: $(δ,\varepsilon)$ are initialized by a closed-form Slifker-Shapiro quantile fit and refined by Bayesian optimization on the validation objective, while the inner training loop is identical to standard RevIN-style training. Across six representative backbones x five real-world datasets x three prediction horizons (90 configurations), decoupled shape optimization recovers $(δ^\star,\varepsilon^\star)$ that sit systematically far from the linear limit, with values that vary in a backbone-dependent way. This empirically supports the central thesis: different backbones genuinely require different normalization parameters to reach their best performance.

Estimating causal effects from observational data requires identifying valid adjustment sets. This task is especially challenging in realistic settings where latent confounding and feedback loops are present. Existing approaches typically assume acyclicity or rely on global causal structure learning, limiting applicability and computational efficiency. In this work, we study a local, data-driven method for covariate selection based on conditional independence information. While this method is known to be sound and complete in acyclic causal models, its validity in the presence of cycles has remained unclear. Our main contribution is to show that these guarantees extend to cyclic causal models. In particular, our result relies on the invariance of conditional independence assertions under $σ$-acyclification. These findings establish a unified, cycle-agnostic perspective on covariate selection and causal effect estimation, showing that the method applies across cyclic and acyclic settings without modification. Empirically, we validate this on extensive synthetic data, showing reliable performance in cyclic causal models.

The proliferation of large-scale and structurally complex data has spurred the integration of machine learning methods into statistical modeling. Recurrent neural networks (RNNs), a foundational class of models for time-dependent data, can be viewed as nonlinear extensions of classical autoregressive moving average models. Despite their flexibility and empirical success in machine learning, RNNs often suffer from limited interpretability and slow training, which hinders their use in statistics. This paper proposes the Parallelized RNN (ParaRNN), a novel model composed of multiple small recurrent units. ParaRNN admits an additive representation that decouples recurrent dynamics into interpretable components, whose behavior can be characterized through recurrence features. This interpretability enables its applications in nonparametric regression for time-dependent data, while the design also allows efficient parallelization. The approximation capacity and non-asymptotic prediction error bounds in a nonparametric regression setting are established for ParaRNN. Empirical results on three sequential modeling tasks further demonstrate that ParaRNN achieves performance comparable to vanilla RNNs while offering improved interpretability and efficiency.

Recent advances in large language models (LLMs) have increasingly relied on reinforcement learning (RL) to improve their reasoning capabilities. Three types of approaches have been widely adopted: The first relies on a deep neural network to estimate the value function of the learning policy in order to reduce the variance of the policy gradient. However, estimating and maintaining such a value network incurs substantial computational and memory overhead. The second avoids training a value network by approximating the value function using sample averages. However, it samples a large number of reasoning traces per prompt for accurate value function approximation, making it computationally expensive. The third samples only a single reasoning trajectory per prompt, which reduces computational cost but suffers from poor sample efficiency. This paper focuses on a practical, resource-constrained setting in which only a small number of reasoning traces can be sampled per prompt, while low-variance gradient estimation remains essential for high-quality policy learning. To address this challenge, we bring classical nonparametric statistical methods, which are both computationally and statistically efficient, to LLM reasoning. We employ kernel smoothing as a concrete example for value function estimation and the subsequent policy optimization. Numerical and theoretical results demonstrate that our proposal achieves accurate value and gradient estimation, leading to improved policy optimization.

Face recognition from a single image per person is a challenging problem because the training sample is extremely small. We consider a variation of this problem. In our problem, we recognize only one person, and there are no labeled data for any other person. This setting naturally arises in authentication on personal computers and mobile devices, and poses additional challenges because it lacks negative examples. We formalize our problem as one-class classification, and propose and analyze an algorithm that learns a non-parametric model of the face from a single labeled image and a stream of unlabeled data. In many domains, for instance when a person interacts with a computer with a camera, unlabeled data are abundant and easy to utilize. This is the first paper that investigates how these data can help in learning better models in the single-image-per-person setting. Our method is evaluated on a dataset of 43 people and we show that these people can be recognized 90% of time at nearly zero false positives. This recall is 25+% higher than the recall of our best performing baseline. Finally, we conduct a comprehensive sensitivity analysis of our algorithm and provide a guideline for setting its parameters in practice.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

In contextual bilateral trade under full feedback, the posted price does not affect which valuations are observed. We show that in this model such action-independent feedback removes the polynomial adaptation penalty familiar from heavy-tailed bandits: fully parameter-free algorithms attain the oracle minimax $T$-exponents up to logarithmic factors, with no knowledge of the moment order $p \in (1,2)$ or its scale $σ_p$, and -- in the nonparametric case -- none of the effective Hölder smoothness $β\in (0,1]$. The statistic that makes model selection possible is a paired squared-loss difference, whose noise-square term cancels exactly, leaving noise damped by the candidate gap. The resulting bilateral-trade regret rates are new. Trader valuations have bounded conditional densities and heavy tails -- finite $p$-th moments for some $p \in (1,2)$, with possibly infinite variance. An epoch-based algorithm with truncated means achieves regret $\widetilde{O}(T^{(2-p)/p})$ in the parametric model and $\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$ when the market value function is $β$-Hölder, with matching $Ω(\cdot)$ lower bounds -- under a mild nondegeneracy condition -- via Assouad's method and a fixed-support mixture construction -- characterizing the minimax rate in $T$ up to logarithmic factors over the effective smoothness range $β\in (0,1]$, interpolating between the classical nonparametric rate at $p{=}2$ and the trivial linear rate as $p \to 1^+$. The enabling structural step extends the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations: within our conditionally independent, conditionally centered noise model, bounded conditional densities and finite first moments suffice for the expected regret of any price $π$ to satisfy $\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$ -- no second moment is needed.

Regression with non-Euclidean responses---e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions---has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses---which are considered as random objects in a metric space---from Euclidean predictors. Our method utilizes the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses, without making model assumptions or relying on local smoothing. We further establish rigorous generalization guarantees for DFNNs and derive corresponding risk bounds, providing, to the best of our knowledge, the first such theoretical results for deep learning regression with metric-space-valued responses. Empirical studies on synthetic distributional and network-valued responses, as well as real-world applications to predicting compositional responses in an Aitchison simplex and spherical responses, demonstrate that DFNNs consistently outperform all existing methods.

This paper considers the estimation of quantiles via a smoothed version of the stochastic gradient descent (SGD) algorithm. By smoothing the score function with a bandwidth tied to the learning rate, we obtain estimates that are monotone in the quantile level at every iteration, while retaining the memory and computational efficiency required for streaming data. We establish non-asymptotic tail probability bounds for the smoothed estimate with and without Polyak-Ruppert averaging, which are sub-exponential with a multi-regime structure. For the averaged estimate we further derive a Bahadur representation that is uniform in the quantile level and across coordinates, and a resulting Gaussian approximation by the maximum of Brownian bridges, with the dimension $p$ allowed to grow exponentially in the sample size. This yields simultaneous inference across coordinates and quantile levels. As an alternative that avoids estimating the sparsity function, we propose an online multiplier bootstrap that preserves monotonicity, runs in a single pass and is asymptotically valid. Extending the theory to a localized recursion, we obtain online nonparametric conditional quantile estimates with uniform bands over design points and quantile levels. Simulations confirm accurate finite-sample coverage, and we illustrate the method on conditional value-at-risk curves.

In the context of multivariate nonparametric regression with missing covariates, we propose Pattern Embedded Neural Networks (PENNs), which can be applied in conjunction with any existing imputation technique. In addition to a neural network trained on the imputed data, PENNs pass the vectors of observation indicators through a second neural network to provide a compact representation. The outputs are then combined in a third neural network to produce final predictions. Our main theoretical result exploits an assumption that the observation patterns can be partitioned into cells on which the Bayes regression function behaves similarly, and belongs to a compositional Hölder class. It provides a finite-sample excess risk bound that holds for an arbitrary missingness mechanism, and in combination with a complementary minimax lower bound, demonstrates that our PENN estimator attains in typical cases the minimax rate of convergence as if the cells of the partition were known in advance, up to a poly-logarithmic factor in the sample size. Numerical experiments on simulated, semi-synthetic and real data confirm that the PENN estimator consistently improves, often dramatically, on standard neural networks without pattern embedding. Code to reproduce our experiments, as well as a tutorial on how to apply our method, is publicly available.

Analyzing the behavior of complex interdependent networks requires complete information about the network topology and the interdependent links across networks. For many applications such as critical infrastructure systems, understanding network interdependencies is crucial to anticipate cascading failures and plan for disruptions. However, data on the topology of individual networks are often publicly unavailable due to privacy and security concerns. Additionally, interdependent links are often only revealed in the aftermath of a disruption as a result of cascading failures. We propose a scalable nonparametric Bayesian approach to reconstruct the topology of interdependent infrastructure networks from observations of cascading failures. Metropolis-Hastings algorithm coupled with the infrastructure-dependent proposal are employed to increase the efficiency of sampling possible graphs. Results of reconstructing a synthetic system of interdependent infrastructure networks demonstrate that the proposed approach outperforms existing methods in both accuracy and computational time. We further apply this approach to reconstruct the topology of one synthetic and two real-world systems of interdependent infrastructure networks, including gas-power-water networks in Shelby County, TN, USA, and an interdependent system of power-water networks in Italy, to demonstrate the general applicability of the approach.