Papers

Unobserved confounding is a fundamental challenge for estimating causal effects. To address unobserved confounding, recent literature has turned to two different approaches – proxy variables and the use of multiple treatments. The first approach, commonly referred to as proximal causal inference, requires proxies to be assigned to specific asymmetric roles: treatment-inducing proxies (negative control exposures), variables that act as common causes of the treatment and outcome, and outcome-inducing proxies (negative control outcomes). In practice, however, identifying variables that satisfy these asymmetric roles can be difficult depending on the application domain. The second approach, commonly referred to as the “Deconfounder," deals with multiple conditionally independent treatments. There has been limited progress towards developing a consistent estimation method for this setting. As the primary contribution of this work, we establish that causal effects are identifiable in both settings when the unobserved confounder is categorical under suitable conditions. Our approach builds on a mixture learning perspective: we show that the underlying confounding structure can be recovered by identifying the corresponding mixture distribution. We propose an estimation procedure based on tensor decomposition, which allows consistent recovery of the latent structure and comes with non-asymptotic guarantees. Simulation studies and real data experiments demonstrate that the proposed method performs well even with limited data.

Heavy-tailed distributions are prevalent in performance evaluation, network traffic, and risk modeling. This behavior poses a fundamental challenge for modern deep generative models. Standard Variational Autoencoders (VAEs) employ Gaussian decoder likelihoods and Lipschitz-constrained neural networks, a combination that is structurally incapable of producing heavy-tailed outputs: the Gaussian tail decays exponentially, and Lipschitz continuity prevents the decoder from amplifying rare events from the latent space input to sufficiently overcome this decay. We provide both a theoretical characterization of this limitation and a controlled empirical demonstration using synthetic Pareto data across a grid of tail indices $\alpha$ $\in$ {2, 3, 5, 30} and dimensions d $\in$ {1, 5, 10}. As a solution, we replace the Gaussian decoder with a Phase-Type (PH) distribution based on Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions allow for arbitrarily precise approximations of any positive-valued distributions, including heavy-tailed families. Experiments showed that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to x6 and extreme quantile error by up to x10 compared to the Gaussian baseline for heavy-tailed data. These results demonstrate that integrating Markov chain-based distributions into the decoder of a generative model institutes a principled and practically effective solution to the heavy-tail generation problem.

Bayesian latent space models offer a principled approach to network representation, but rely on correct specification of both geometry and link function. Real-world networks often violate these assumptions, exhibiting geometric mismatch and structural anomalies that break standard metric properties. We show that such misspecification pushes the data-generating distribution outside the model class, causing Bayesian inference to become overconfident and poorly calibrated. To address this, we propose a generalized posterior framework for random geometric graphs. We introduce Link-Sequential R-SafeBayes, a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization. Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.

We develop kernel criteria for the likelihood-ratio, hazard-rate, usual stochastic, and relative log-concavity orders in parametric families of univariate probability laws with densities. The score is the derivative of the log density with respect to the parameter, and a kernel equals the score up to an additive term depending only on the parameter. Kernel monotonicity gives likelihood-ratio order, kernel concavity gives relative log-concavity, and two tail-conditional mean inequalities give the hazard-rate and usual stochastic orders. The same construction applies along joint-parameter paths and to comparisons between two laws whose densities admit parameter-dependent factors, where the log-factor ratio is used as the kernel. For compound sums with a random number of i.i.d. terms, the induced kernel is the posterior mean of the kernel of the summand count. The applications recover standard one-parameter orderings, give likelihood-ratio comparisons for compound laws, and handle nonmonotone examples through the tail-conditional criteria.

Data assimilation (DA) addresses the problem of sequentially estimating the state of a dynamical system from noisy and incomplete observations. In this work, we employ a diffusion model as a world model to simulate and predict the system's dynamics. Recently, score-based diffusion models have learned global diffusion priors that effectively model (stochastic) dynamics, revealing strong potential for data assimilation. In this paper, we investigate how information from noisy observations can be incorporated to enable continuous correction and refinement of the predicted system state when using a diffusion prior. Motivated by particle filtering methods, we represent the posterior distribution using a set of particles. After receiving noisy observations, the diffusion model is guided using the observation likelihood to steer the generation process toward observation-consistent states. Nevertheless, such guidance does not guarantee sampling from the true posterior. We therefore employ a Sequential Monte Carlo approach over the diffusion trajectory, viewed as a path measure, to reweight and resample particles, thereby correcting the generation process and ensuring convergence toward the desired posterior distribution. This leads to an unbiased particle filtering method that rigorously fuses observational data with diffusion model simulations.

When a statistical model $\{P_{\theta} : \theta \in \Theta\}$ lacks analytically tractable likelihoods, parametric statistical inference based on data generated from an unknown underlying distribution $P$ can still be performed as long as simulations from the model are possible. This approach is called Simulation Based Inference (SBI). Statistical models are rarely exactly correct (that is, $P \notin \{P_{\theta}: \theta \in \Theta\}$), and Robust SBI focuses on inferring a reasonable parameter even under model mis-specification. We focus on the setting where $P$ possesses potentially both geometric and Total Variation type discrepancies from $P_{\theta^*}$. For this problem, we use a Kullback-Liebler informed robust Optimal Transport divergence, motivated by Empirical Likelihood considerations. We introduce a stochastic sub-gradient ascent algorithm with a convergence guarantee for estimating the semi-discrete version of this robust Optimal Transport divergence, and design a parallelized SBI algorithm which employs the regular bootstrap on top of minimum semi-discrete robust Optimal Transport for parameter uncertainty quantification. We demonstrate mathematically why the divergence is robust under a joint geometric plus Total Variation type contamination and then illustrate the robustness of inferences on a complex benchmark SBI task.

Numerous studies have shown that microbial metabolites, which represent the products of bacteria in the human gut, play a key role in shaping cancer risk and response to treatment. However, metabolite data typically contain a large proportion of missing values, which may result from either low abundance or technical challenges in data processing. Moreover, given the compositionality of microbiome data, where the observed abundances can only be interpreted on a relative scale, standard variable selection methods are not applicable. In this project, we propose a novel Bayesian regression method to address these challenges in the integration of metabolite and microbiome data. Key features of our proposed model include modeling the two different mechanisms of missingness for the metabolite data and adopting a Bayesian prior designed to address the compositional characteristics of microbiome data. We demonstrate on simulated data that our proposed model can accurately impute the unobserved true metabolite values and correctly select the relevant microbiome predictors. We further illustrate our method using real data from a study focused on understanding the interplay between the microbiome and metabolome in colorectal cancer.

Causal mediation analysis decomposes the total treatment effect into a portion operating through a hypothesized mediator and a residual direct portion. Identification of natural direct and indirect effects typically rests on the mediator stage of sequential ignorability, which cannot be empirically verified and requires explicit sensitivity analysis. We formulate the bridge score, a mediator-stage balancing score, as a low-dimensional vector formed from the two treatment-specific mediator densities at a common mediator value, and show that it balances baseline covariates for the mediator stage relevant to natural effect identification. Conditional on the bridge score, we derive a sharp pointwise variance envelope on the unidentified mediator-outcome confounding function in terms of latent outcome relevance and residual selection. To make the bound operational for sensitivity analysis, we further introduce a residual budget calibration approach based on local residual outcome variation and record a complementary range bound for support-based restrictions. Finally, we show how the pointwise bound can be operationalized for inference through a scalar functional reduction and a Bayesian g-computation algorithm that combines observed-data posterior uncertainty with user-specified sensitivity uncertainty, rather than treating the unidentified sensitivity corrections as learned from the likelihood.

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular $\mathtt{Adam}$ and $\mathtt{AdamW}$, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case, $\mathtt{AdaGrad}$, the origin of adaptive gradient methods. We provide the first provable convergence rate for $\mathtt{AdaGrad}$ in non-convex optimization when the tail index $p$ satisfies $4/3<p\leq2$. Notably, this result is achieved without requiring any prior knowledge of $p$ and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by $\mathtt{AdaGrad}$. Lastly, we consider $\mathtt{AdaGrad}\text{-}\mathtt{Norm}$, a popular variant of $\mathtt{AdaGrad}$ in theoretical studies, and show an improved rate that holds for any $1<p\leq2$ under an extra mild assumption.

The Central Limit Theorem provides a foundation for inferential statistics and hypothesis testing. It describes how standardized statistics behave under repeated sampling from large populations. However, if the size of the sample (n) becomes so large that it approaches the size of the population (N), sampling variability becomes very small, and standard errors and margins of error both approach zero. The purpose of this project was to investigate the behavior of estimators as the sampling fraction (f = n/N) approaches 1, motivated by modern data streams from administrative records, transaction logs, sensor systems, and institutional databases that capture large portions of finite populations. We constructed two finite populations with known parameters and drew repeated samples across a range of sampling fractions. We then examined the resulting randomization distributions of the sample mean to understand how sampling variability collapses. Additional experiments were conducted using various CPU- and GPU-based methods to evaluate the deviation of the sample mean from the defined population mean under different computational conditions. The results confirm that sampling variability diminishes as expected under finite population theory and becomes negligible well before full enumeration is reached. Once sampling variability is minimized, remaining deviations in estimators are primarily related to numerical precision and computational structure rather than random sampling. These findings support a reassessment of inferential assumptions in high-coverage, large-scale data settings.

Federated Learning is a leading framework for training ML and AI models collaboratively across numerous user devices or databases. We study the trade-offs among estimation accuracy, privacy constraints, and communication cost for differentially private (DP) federated M estimation. The two standard methods in the literature are FedAvg, which may suffer from high federation bias, and FedSGD, which can incur high communication cost. Aimed at improving accuracy at a reduced communication cost, we propose FedHybrid, which uses FedSGD starting with an improved initialization by the FedAvg estimator. We propose FedNewton, which averages local Newton iterations to reduce bias in FedAvg, achieving an estimation accuracy comparable to FedSGD with much fewer communication rounds when the number of clients grows sufficiently slowly. We establish finite sample upper bounds on the mean-squared error rates of the DP versions of these estimators as functions of the number of clients, local sample sizes, privacy budget, and number of iterations. We further derive a minimax lower bound on the MSE of any iterative private federated procedure that provides a benchmark to assess the optimality gap of these methods. We numerically evaluate our methods for training a logistic regression and a neural network on the computer vision datasets MNIST and CIFAR-10.

Inverse problems are ubiquitous in modern scientific studies and involve recovering an underlying signal from noisy observations often transformed by a measurement operator. These problems are frequently ill-posed, particularly in imaging, leading to multiple plausible solutions and considerable uncertainty in reconstructed images. In fields like the physical and biological sciences, accurate uncertainty quantification (UQ) is critical for trustworthy scientific analyses and confident diagnoses. Current UQ methods for imaging often fall short; they can be inaccurate, or require unavailable or difficult-to-acquire ground truth data for calibration, which can introduce hidden biases due to distribution shifts between calibration and observed data. We introduce a UQ approach that leverages equivariant bootstrapping to generate heuristic coverages by exploiting data symmetries. We then refine these coverages through a conformal prediction calibration step, while crucially employing a self-supervised approach to avoid the need for ground truth calibration data. We demonstrate this method with weak lensing mass-mapping, where we aim to reconstruct the convergence field from shear measurements of distant galaxies weakly-lensed by gravitational fields. Mass-mapping in particular benefits from the self-supervised approach, as simulating calibration data is expensive and relies on specific cosmological models that could introduce biases in downstream cosmological inference tasks.

Directed Acyclic Graphs (DAGs) are central to uncovering causal structure in complex systems, yet learning a single DAG from data is often challenging: model uncertainty, finite samples, and a combinatorially large search space frequently yield unstable estimates. We propose DAGgr, a model averaging framework that aggregates multiple candidate DAGs into a single stable representation. Candidate graphs are weighted by their out-of-sample predictive likelihood across repeated data splits, and a thresholding rule on the resulting edge-importance scores guarantees that the aggregated graph is itself acyclic. We establish a finite-sample risk bound, prove that the procedure preserves acyclicity, and show that edge selection is consistent under mild conditions on the weights. Simulations across random, hub, and chain structures, together with an analysis of the Sachs et al. (2005) protein-signaling network, show that DAGgr matches or exceeds the best individual candidate while consistently outperforming bootstrap-aggregation baselines across structural recovery metrics.

Markov random fields are common prior distributions used in Bayesian inverse imaging problems. In particular, difference priors assign probability distributions to differences between neighbouring pixels, such as Gaussian, Laplace, or Cauchy distributions. Depending on the chosen difference distribution, these priors have smoothing or edge-preserving properties. In this work, we propose a hyperprior on the connectivity graph of the pixel grid in the form of a random spanning tree, i.e., a random connected graph with the minimal number of edges, thereby coupling continuous and discrete random variables in the prior. By using random spanning trees, only a sparse random subset of edges is regularized, which helps preserve edges in the image with reduced contrast loss compared to standard difference-based Markov random fields. We discuss how fractal-like interfaces arise in high-resolution prior samples due to the random-tree connectivity. Finally, we propose a Gibbs sampler that alternates between the discrete tree updates and continuous pixel updates to efficiently explore the posterior distribution. We apply the method to various standard test image restoration problems, including denoising, deblurring, and inpainting, to study the impact of the proposed prior in comparison with existing Markov random fields.

We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.

We present the Open-Source Sleep Monitor and Modulator (OSSMM), an open-source hardware and software platform for accessible sleep research. The OSSMM comprises a small wearable headband built from 3D prints and affordable commercial-off-the-shelf (COTS) components at a material cost under 40 euros, supported by a companion Android application. The system requires no conductive gels, disposable electrodes, or specialized equipment, and captures multiple biosignals movement, pulse, electrooculography (EOG), and putative electroencephalography (EEG) with wireless connectivity for data storage and potential sleep modulation capability via an onboard vibration motor. A proof-of-concept single-participant evaluation across 15 nights demonstrated that the captured biosignals support four-stage sleep classification (Wake, Light Sleep, Deep Sleep, REM) using conventional machine learning methods, with the best-performing model achieving a Macro F1-score of 0.770 and accuracy of 0.776 against a validated non-contact sleep monitor ($\kappa$=0.63 with PSG). Two technical findings are of particular note. First, inexpensive, reusable conductive thermoplastic polyurethane (CTPU) electrodes from commercial fitness chest straps captured a differential signal whose spectral properties in canonical EEG frequency bands, including signatures consistent with sleep spindles, are the principal features driving classification. Second, this signal is obtained from just two frontal electrodes without a dedicated ground reference, suggesting that practical sleep staging is achievable with simpler configurations than typically employed. All hardware designs, software, and build instructions are openly available to support replication and modification by the research community.

Accurate estimates of item difficulty are essential for valid assessment and effective adaptive learning. However, for newly created tasks, response data are typically unavailable. Pretesting and expert judgement can be costly and slow, while machine learning methods often require large labelled training datasets. Recent work suggests that large language models (LLMs) may help. However, there is limited evidence on the elicitation procedures and prompt configurations used to emulate experts for difficulty estimation. This study addresses this gap by evaluating three off-the-shelf LLMs as difficulty raters for newly created items without access to response data. Using an item bank from an online learning system, the study examined 6 domains of primary-school mathematics, with empirical difficulty estimates treated as empirical reference. The study used a full factorial design crossing three factors: judgement format (absolute vs pairwise), decision type (hard decisions vs token-probability-based estimates), and prompting strategy (zero-shot vs few-shot). LLM-derived difficulty estimates were compared with empirical difficulties using Spearman rank correlations. Across domains, LLM-based estimates exhibited moderate to strong positive correlations with empirical item difficulties. For simpler arithmetic tasks, some configurations approached the upper end of the accuracy range reported for human experts in previous research. Pairwise comparison consistently outperformed absolute judgement in the absence of additional refinements. However, when token-level probabilities were incorporated and examples of items with known empirical difficulty were provided, the absolute judgement configuration likewise demonstrated moderate-to-high alignment. The study positions LLMs as a promising tool for initial item calibration and offers insights into effective workflow configuration.

Federated Bayesian neural networks require fixing a prior on the model parameters, which is notoriously difficult, and misspecification of this prior can severely degrade accuracy and calibration. Motivated by the rapid progress of predictive models, the martingale posterior, also known as predictive Bayes, replaces the prior–likelihood pair with a predictive distribution and recovers parameter uncertainty by repeatedly drawing predictive samples and refitting the model. This letter proposes {federated martingale posterior} (FMP) sampling, a one-shot embarrassingly parallel protocol in which each client uploads a small set of trainable data embeddings and the server runs the predictive sampler centrally. Analysis of the sampling error demonstrates the impact of the dataset compression rate, while experiments show that FMP closely matches the centralized counterpart and achieves the lowest mean expected calibration error (ECE) among the evaluated one-shot federated methods.

This paper introduces the Manifold Probe, a supervised method for discovering representation manifolds in superposition. The method generalizes linear regression probes by learning the space of features of a concept that can be linearly predicted from the representations, and then learning the directions used to encode them. We demonstrate the probe on representations of time and space in Llama 2-7b, finding manifolds which linearly represent an interpretable set of features in each case. In the case of time, we show that by steering along the manifold, we can influence the model's completions about the years in which famous songs, movies and books were released, providing evidence that the Manifold Probe can discover manifolds which are causally involved in model behaviour.

While diffusion has drawn considerable recent attention from the language modeling community, continuous diffusion has appeared less scalable than discrete approaches. To challenge this belief we revisit Plaid, a likelihood-based continuous diffusion language model (DLM), and construct RePlaid by aligning the architecture of Plaid with modern discrete DLMs. In this unified setting, we establish the first scaling law for continuous DLMs that rivals discrete DLMs: RePlaid exhibits a compute gap of only $20\times$ compared to autoregressive models, outperforms Duo while using fewer parameters, and outperforms MDLM in the over-trained regime. We benchmark RePlaid against recent continuous DLMs: on OpenWebText, RePlaid achieves a new state-of-the-art PPL bound of $22.1$ among continuous DLMs and superior generation quality. These results suggest that continuous diffusion, when trained via likelihood, is a highly competitive and scalable alternative to discrete DLMs. Moreover, we offer theoretical insights to understand the advantage of likelihood-based training. We show that optimizing the noise schedule to minimize the ELBO's variance naturally yields linear cross-entropy (information loss) over time. This evenly distributes denoising difficulty without any case-specific time reparameterization. In addition, we find that optimizing embeddings via likelihood creates structured geometries and drives the most significant likelihood gain.

We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel $K_q(x,y) = -|x-y|^q$, $q \in (0,2)$. The resulting discrepancy is the classical energy distance $$\mathcal E_q^2(\mu, \omega) = -\frac{1}{2}\iint_{\mathbb{R}^d \times \mathbb{R}^d} |x-y|^q \, d(\mu - \omega)(x)\, d(\mu - \omega)(y),$$ and we ask how fast the best $N$-point empirical approximation $\inf_{\mu_N \in \mathcal{P}^N}\mathcal{E}_q(\mu_N,\omega)$ decays as $N \to \infty$. Given a probability measure $\omega$ on $\mathbb{R}^d$ with compact support satisfying an Ahlfors regularity condition of exponent $\beta \in (0,d]$, we prove that the sharp two-sided bound $$\mathcal E_q(\mu_N, \omega) \asymp N^{-\frac{1}{2}\left(1 + \frac{q}{\beta}\right)}$$ holds both for the worst-case empirical measure $\mu_N$ (lower bound, holding for every configuration of $N$ points) and for an optimally chosen empirical measure $\mu_N$ (upper bound). This complements the qualitative consistency result of Fornasier and Hütter \cite{fornasier2014consistency}, who proved narrow convergence of the minimizers of $\mathcal E_q^2(\cdot, \omega)$ over empirical measures without quantitative rates.

Coding and computation remain major bottlenecks in Markov chain Monte Carlo (MCMC) workflows, especially as modern sampling algorithms have become increasingly complex and existing probabilistic programming systems remain limited in model support, extensibility, and composability. We introduce AI4BayesCode, an extensible LLM-driven system that translates natural-language Bayesian model descriptions into runnable, validated MCMC samplers. To improve reliability, AI4BayesCode adopts a modular design that decomposes models into modular sampling blocks and maps each block to a built-in sampling component, reducing the need to implement complex sampling algorithms from scratch. Reliability is further improved through pre-generation validation of model specifications and post-generation validation of generated sampler code. AI4BayesCode also introduces a novel recursively stateful coding paradigm for MCMC, allowing modular sampling components, potentially developed by different contributors, to be composed coherently within larger MCMC procedures. We develop a benchmark suite to evaluate AI4BayesCode for sampler-generation. Experiments show that AI4BayesCode can implement a wide range of Bayesian models from natural-language descriptions alone. As an open-ended system, its capability can continue to expand with improvements in the underlying AI agent and the addition of new built-in blocks. The software is available at https://ai4bayescode.com/.

Generative models can produce nonsensical text, unrealistic images, and unstable materials faster than simulation or human review can absorb; without per-sample confidence, trust erodes. Existing fixes run $k$ ensembles or stochastic trajectories at $k\times$ compute, measuring variability between models, not model confidence. We propose Flow Matching with Confidence (FMwC). FMwC injects input-dependent multiplicative noise at selected layers, propagates its variance through the network in closed form, and integrates it along the ODE trajectory, yielding a per-sample confidence score at standard sampling cost. The score supports multiple uses: filtering improves image quality and thermodynamic stability of crystals; editing rewinds trajectories to the points where the model commits and redirects them; and adaptive stepping concentrates ODE compute where the flow is ambiguous. We find that the confidence score correlates with the magnitude of the divergence of the learned velocity field, which gives us a window to understand the generative process, opening up surgical forms of guidance that target the moments that matter, new sampling algorithms and interpretability of generative models.

This paper studies approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max\{0,t\}^s$, together with their generalization behavior under $\ell_1$ path-norm control. For the $L^p$-type integral spaces $\widetilde{\mathcal{F}}_{p,\tau_d,s}$, $1\le p\le2$, spherical harmonic analysis yields approximation bounds for shallow networks. In particular, when $\tau_d$ is the uniform measure and $1\le p<2$, the approximation rate is $O\!\left(m^{-\frac{p(2s+2d+1)-2d}{2dp}}\right)$ for $1\le p\le p^*$ and $O\!\left(m^{-\frac{p(4s+3d-1)-2d+2}{4dp}}\right)$ for $p^*<p<2$, where $p^*=\frac{2d+2}{d+3}$. Approximation bounds for Sobolev spaces $W^{\alpha,p}$, $1\le p<2$, are obtained through embeddings into spectral Barron spaces. For nonparametric regression with sub-Gaussian noise, path-norm-regularized shallow ReLU$^s$ networks achieve minimax-optimal rates $O\!\left(n^{-\frac{d+2s+1}{2d+2s+1}}\log n\right)$ over $\mathscr{B}_s$ and $O\!\left(n^{-\frac{2\alpha}{2\alpha+d}}\log n\right)$ over $W^{\alpha,\infty}$, with matching lower bounds up to logarithmic factors.

Adaptive experimentation enables efficient estimation of causal effects, but existing methods are not designed for survival data with censoring, where event times are only partially observed (e.g., overall survival in cancer trials but with dropout). In this paper, we develop a novel framework for adaptive experimentation to estimate causal effects under right censoring. For this, we derive the semiparametric efficiency bound for the average survival effect curve as a function of the treatment allocation policy and thereby obtain a closed-form efficiency-optimal allocation policy. The policy generalizes classical Neyman allocation to survival settings by prioritizing patient strata where both event and censoring dynamics induce high uncertainty. Building on this, we propose the Adaptive Survival Estimator (ASE), an adaptive framework that learns the allocation policy and estimates the average survival effect curve sequentially. Our framework has three main benefits: (i) it accommodates arbitrary machine learning models for nuisance estimation; (ii) it is guided by a closed-form efficiency-optimal allocation policy; and (iii) it admits strong theoretical guarantees, including asymptotic normality via a martingale central limit theorem. We demonstrate our framework across various numerical experiments to show consistent efficiency gains over uniform randomization and censoring-agnostic baselines.

We develop asymptotic theory for principal component analysis (PCA) of a high-dimensional factor model in which the working dimension $R$ is fixed and only required to satisfy $R \ge r$, where $r$ is the true number of factors. Building on anisotropic local laws from random matrix theory, we show that the “extra” empirical eigencomponents beyond the $r$-th are asymptotically noise-governed, incoherent, and nearly orthogonal to the factor loadings. We introduce two rotations, an expanded $r\times R$ map $H'$ and a compressed $R\times r$ map $H^{+}$, and establish consistency of the estimated factors under both. As an application, we analyze a factor-augmented regression for treatment-effect inference and prove $\sqrt{T}$-asymptotic normality for every fixed $R \ge r$. These results provide a theoretical underpinning for the common empirical practice of adopting a conservative upper bound on the number of factors, and shift the analytical burden from consistent dimension selection to the milder requirement of bounding $r$ from above.

Physical AI is being successfully applied to data which does not follow the traditional paradigm of independent and identically distributed (i.i.d.) samples. In fact, physical AI is often trained on data which is not random at all, and is instead derived from chaotic dynamical systems like turbulence. We aim to explain the empirical success of these methods using the example of generative adversarial networks (GANs), whose statistical learning theory under the i.i.d. assumption is generally well understood. We prove that it is possible, using an infinite-dimensional model of generative adversarial learning (GAL), to learn the invariant distribution of a sufficiently chaotic dynamical system from a single deterministically evolving time series of its states or measurements thereof, and give explicit rates for the convergence to the solution in terms of the Jensen-Shannon divergence.

The functional ANOVA provides a fundamental representation of square-integrable multivariate functions into main effects and higher-order interactions. For independent inputs, the components belong to mutually orthogonal Hilbert subspaces and admit an explicit representation. For dependent inputs, however, the components are only hierarchically orthogonal: although existence and uniqueness results are available, the Hilbert subspaces underlying the generalized decomposition have remained implicit. We resolve this representation problem for continuous inputs supported on a bounded hyperrectangle whose joint density is bounded above and away from zero. We introduce a distribution-adapted family of functions and prove that it forms a Riesz basis of the $L^2$ space, thereby guaranteeing a unique, stable, and unconditionally convergent representation. We then show that, for every coalition of variables, the corresponding block of this basis exactly characterizes the Hilbert subspace containing the functional ANOVA component. Our construction recovers the classical orthogonal decomposition under input independence. As a direct consequence, computing the generalized functional ANOVA reduces to estimating coefficients in an explicit, distribution-adapted basis. Finally, as a proof of concept, we introduce an elementary, fast and model-agnostic estimator based on our theoretical results. Experiments on synthetic and real-world datasets illustrate its connections with established tabular explanation methods and show that low order components often capture most of the signal in the model output.

The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [arXiv:2603.04525] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity $O(J^2)$ in the number of time steps $J$, an FFT-based acceleration with complexity $O(J\log J)$ for convolution kernels on uniform grids, and an exact recursion with complexity $O(JR^2)$ for kernels admitting a state-space representation of dimension $R$; retaining standard signature complexity in the path dimension and truncation level $N$. We further show that the number of factors in matrix-valued kernels of the form $K(t,s)=\sum_p k_p(t-s)A_p$ do not increase the asymptotic complexity in $J$ and $N$. Finally, we derive a finite-difference predictor–corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package "tensordev".

We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \cite{bahadur1966}. Empirical-process theory provides a usable scaffolding through the mechanics of half-spaces, symmetric differences, and Glivenko–Cantelli uniform convergence. They yield stability bounds, but absorb changes in projection direction and changes in quantile threshold into a single symmetric-difference measure. Interestingly, a global uniform-convergence requirement is imposed on what is intrinsically a local quantile-stability problem. This paper introduces a Q-Q orthogonality formulation for separating projection-direction and quantile-threshold effects. The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms, $\hat q_{\alpha}(\hat w)-q_{\alpha}(w_0)=D_1+D_2+D_3$. Here, $D_1$ measures the population quantile movement induced by perturbing the projection direction, $D_2$ measures the empirical quantile fluctuation with the projection direction held fixed, and $D_3$ is the Bahadur-type remainder.