Papers

We study supervised multiclass classification for diffusion processes, where each class is characterized by a distinct drift function and trajectories are observed at discrete times. We first derive a multidimensional Bayes rule and then construct a plug-in classifier by estimating the class-specific drifts with neural networks. Under standard regularity assumptions, we establish convergence rates for the excess misclassification risk, making explicit the contributions of drift estimation, time discretization, and dimension. Our analysis also highlights the benefit of exploiting the diffusion structure: the drift is learned from all observed increments, leading to sharper guarantees than direct trajectory-based neural classifiers in the considered setting. Numerical experiments support the theory: the proposed method achieves better classification performance than Denis et al. (2024) in dimension one, remains effective in higher dimensions when the drift functions admit a compositional structure, and outperforms end-to-end neural classifiers trained directly on trajectories, as in Bos & Schmidt-Hieber (2022).

Spatial interference and spatial confounding are two major issues inhibiting precise causal estimates when dealing with observational spatial data. Moreover, the definition and interpretation of spatial confounding remain arguable in the literature. In this paper, our goal is to provide clarity in a novel way on misconception and issues around spatial confounding from Directed Acyclic Graph (DAG) perspective and to disentangle both direct, indirect spatial confounding and spatial interference based on bias induced on causal estimates. Also, existing analyses of spatial confounding bias typically rely on Normality assumptions for treatments and confounders, assumptions that are often violated in practice. Relaxing these assumptions, we derive analytical expressions for spatial confounding bias under more general distributional settings using Poisson as example . We showed that the choice of spatial weights, the distribution of the treatment, and the magnitude of interference critically determine the extent of bias due to spatial interference. We further demonstrate that direct and indirect spatial confounding can be disentangled, with both the weight matrix and the nature of exposure playing central roles in determining the magnitude of indirect bias. Theoretical results are supported by simulation studies and an application to real-world spatial data. In future, parametric frameworks for concomitantly adjusting for spatial interference, direct and indirect spatial confounding for both direct and mediated effects estimation will be developed.

This paper presents a focused review of Markov random fields (MRFs)–commonly used probabilistic representations of spatial dependence in discrete spatial domains–for categorical data, with an emphasis on models for binary-valued observations or latent variables. We examine core structural properties of these models, including clique factorization, conditional independence, and the role of neighborhood structures. We also discuss the phenomenon of phase transition and its implications for statistical model specification and inference. A central contribution of this review is the use of response functions, a unifying tool we introduce for prior analysis that provides insight into how different formulations of MRFs influence implied marginal and joint distributions. We illustrate these concepts through a case study of direct-data MRF models with covariates, highlighting how different formulations encode dependence. While our focus is on binary fields, the principles outlined here extend naturally to more complex categorical MRFs and we draw connections to these higher-dimensional modeling scenarios. This review provides both theoretical grounding and practical tools for interpreting and extending MRF-based models.

Despite the ubiquity of multiway data across scientific domains, there are few performant and user-friendly methods that fit non-standard nonnegative tensor factorization models tailored to the data at-hand. Researchers may use gradient-based automatic differentiation, which often struggles under nonnegative constraints, choose between a limited set of methods with mature implementations, or implement their own model from scratch. As an alternative, we introduce NNEinFact, an einsum-based multiplicative update algorithm that fits any nonnegative tensor factorization expressible as a tensor contraction by minimizing one of many user-specified loss functions, including the $(\alpha,\beta)$-divergence. To use NNEinFact, the researcher specifies their model with a string. NNEinFact converges to a stationary point of the loss, supports missing data, and fits to tensors with hundreds of millions of entries in seconds. Empirically, NNEinFact fits custom models which outperform standard ones in prediction tasks on real-world tensor data by over $37\%$ and attains less than half the test loss of gradient-based methods while converging up to 90 times faster. Software is publicly available at https://github.com/jhood3/einfact.

Functional ANOVA provides a nonparametric modeling framework for multivariate covariates, enabling flexible estimation and interpretation of effect functions such as main effects and interaction effects. However, effect-wise inference in such models remains challenging. Existing methods focus primarily on inference for entire functions rather than individual effects. Methods addressing effect-wise inference face substantial limitations: the inability to accommodate interactions, a lack of rigorous theoretical foundations, or restriction to pointwise inference. To address these limitations, we develop a unified framework for effect-wise inference in smoothing spline ANOVA on a subspace of tensor product Sobolev space. For each effect function, we establish rates of convergence, pointwise confidence intervals, and a Wald-type test for whether the effect is zero, with power achieving the minimax distinguishable rate up to a logarithmic factor. Main effects achieve the optimal univariate rates, and interactions achieve optimal rates up to logarithmic factors. The theoretical foundation relies on an orthogonality decomposition of effect subspaces, which enables the extension of the functional Bahadur representation framework to effect-wise inference in smoothing spline ANOVA with interactions. Simulation studies and real-data application to the Colorado temperature dataset demonstrate superior performance compared to existing methods.

The Electric Propulsion Electrostatic Analyzer Experiment (ÈPÈE) is a compact ion energy bandpass filter deployed on the International Space Station (ISS) in March 2023 and providing continuous measurements through April 2024. This period coincides with the Solar Cycle 25 maximum, capturing unique observations of solar activity extremes in the mid- to low-latitude regions of the topside ionosphere. From these in situ spectra we derive plasma parameters that inform space-weather impacts on satellite navigation and radio communication. We present a statistical processing pipeline for ÈPÈE that (i) estimates the instrument noise floor, (ii) accounts for irregular temporal sampling, and (iii) extracts ionospheric signals. Rather than discarding noisy data, the method learns a baseline noise model and fits the measurement surface using a scaled Vecchia Gaussian process approximation, recovering values typically rejected by thresholding. The resulting products increase data coverage and enable noise-assisted monitoring of ionospheric variability.

Multi-regional clinical trials (MRCTs) are central to global drug development, enabling evaluation of treatment effects across diverse populations. A key challenge is valid and efficient inference for a region-specific estimand when the target region is small and differs from auxiliary regions in baseline covariates or unmeasured factors. We adopt an estimand-based framework and focus on the region-specific average treatment effect (RSATE) in a prespecified target region, which is directly relevant to local regulatory decision-making. Cross-region differences can induce covariate shift, covariate mismatch, and outcome drift, potentially biasing information borrowing and invalidating RSATE inference. To address these issues, we develop a unified causal inference framework with selective information borrowing. First, we introduce an inverse-variance weighting estimator that combines a "small-sample, rich-covariate" target-only estimator with a "large-sample, limited-covariate" full-borrowing doubly robust estimator, maximizing efficiency under no outcome drift. Second, to accommodate outcome drift, we apply conformal prediction to assess patient-level comparability and adaptively select auxiliary-region patients for borrowing. Third, to ensure rigorous finite-sample inference, we employ a conditional randomization test with exact, model-free, selection-aware type I error control. Simulation studies show the proposed estimator improves efficiency, yielding 10-50% reductions in mean squared error and higher power relative to no-borrowing and full-borrowing approaches, while maintaining valid inference across diverse scenarios. An application to the POWER trial further demonstrates improved precision for RSATE estimation.

Often, constraints arise in deployment settings where even lightweight parameter updates e.g. parameter-efficient fine-tuning could induce model shift or tuning instability. We study test-time adaptation of foundation models for few-shot classification under a completely frozen-model regime, where additionally, no upstream data are accessible. We propose arguably the first training-free inference method that adapts predictions to the new task by performing a change of measure over the latent embedding distribution induced by the encoder. Using task-similarity scores derived from a small labeled support set, exponential tilting reweights latent distributions in a KL-optimal manner without modifying model parameters. Empirically, the method consistently competes with parameter-update-based methods across multiple benchmarks and shot regimes, while operating under strictly and universally stronger constraints. These results demonstrate the viability of inference-level distributional correction for test-time adaptation even with a fully-frozen model pipeline.

Adversarial training attains strong empirical robustness to specific adversarial attacks by training on concrete adversarial perturbations, but it produces neural networks that are not amenable to strong robustness certificates through neural network verification. On the other hand, earlier certified training schemes directly train on bounds from network relaxations to obtain models that are certifiably robust, but display sub-par standard performance. Recent work has shown that state-of-the-art trade-offs between certified robustness and standard performance can be obtained through a family of losses combining adversarial outputs and neural network bounds. Nevertheless, differently from empirical robustness, verifiability still comes at a significant cost in standard performance. In this work, we propose to leverage empirically-robust teachers to improve the performance of certifiably-robust models through knowledge distillation. Using a versatile feature-space distillation objective, we show that distillation from adversarially-trained teachers consistently improves on the state-of-the-art in certified training for ReLU networks across a series of robust computer vision benchmarks.

Efron et al. (2004) introduced least angle regression (LAR) as an algorithm for linear predictions, intended as an alternative to forward selection with connections to penalized regression. However, LAR has remained somewhat of a "black box," where some basic behavioral properties of LAR output are not well understood, including an appropriate termination point for the algorithm. We provide a novel framework for inference with LAR, which also allows LAR to be understood from new perspectives with several newly developed mathematical properties. The LAR algorithm at a data level can viewed as estimating a population counterpart "path" that organizes a response mean along regressor variables which are ordered according to a decreasing series of population "correlation" parameters; such parameters are shown to have meaningful interpretations for explaining variable contributions whereby zero correlations denote unimportant variables. In the output of LAR, estimates of all non-zero population correlations turn out to have independent normal distributions for use in inference, while estimates of zero-valued population correlations have a certain non-normal joint distribution. These properties help to provide a formal rule for stopping the LAR algorithm. While the standard bootstrap for regression can fail for LAR, a modified bootstrap provides a practical and formally justified tool for interpreting the entrance of variables and quantifying uncertainty in estimation. The LAR inference method is studied through simulation and illustrated with data examples.

Stochastic approximation methods such as stochastic gradient descent and temporal-difference learning return either the last iterate or the Polyak–Ruppert average. We establish finite-sample Wasserstein-$p$ Gaussian approximation bounds for both outputs under a common framework combining moment bounds for the normalized iterates with quantitative Gaussian approximation of the appropriately weighted driving noise. These bounds yield convergence of admissible moments and high probability bounds with Gaussian leading terms: heavier dependence on the confidence level is confined to a correction that vanishes with the sample size. We verify the framework's conditions for nonlinear stochastic approximation with independent or Markovian observations and linear stochastic approximation with Markovian observations, with applications to stochastic gradient descent and temporal-difference learning.

Bayesian optimization (BO) is a popular, sample-efficient technique for expensive, black-box optimization. One such problem arising in manufacturing is that of maximizing the reliability, or equivalently minimizing the probability of a failure, of a design which is subject to random perturbations - a problem that can involve extremely rare failures ($P_\mathrm{fail} = 10^{-6}-10^{-8}$). In this work, we propose two BO methods based on Thompson sampling and knowledge gradient, the latter approximating the one-step Bayes-optimal policy for minimizing the logarithm of the failure probability. Both methods incorporate importance sampling to target extremely small failure probabilities. Empirical results show the proposed methods outperform existing methods in both extreme and non-extreme regimes.

It is folklore that reusing training data more than once can improve the statistical efficiency of gradient-based learning. While this phenomenon has been extensively studied in linear regression, the benefit of multi-pass gradient descent (GD, which reuses all the data) over one-pass stochastic gradient descent (online SGD, which uses each data point only once) is not well-understood in nonlinear and non-convex settings, except for a loss modification mechanism achieved by the first two passes on the data. In this work, we consider learning a $d$-dimensional single-index model with a quadratic activation, for which it is known that one-pass SGD requires $n\gtrsim d\log d$ samples to achieve weak recovery. We first show that this $\log d$ factor in the sample complexity persists for full-batch spherical GD on the correlation loss; however, by simply truncating the activation, full-batch GD exhibits a favorable optimization landscape at $n \simeq d$ samples, thereby outperforming one-pass SGD (with the same activation) in statistical efficiency. We complement this result with a trajectory analysis of full-batch GD on the squared loss from small initialization, showing that $n \gtrsim d$ samples and $T \gtrsim\log d$ gradient steps suffice to achieve strong (exact) recovery.

Data-driven algorithm design automates hyperparameter tuning, but its statistical foundations remain limited because model performance can depend on hyperparameters in implicit and highly non-smooth ways. Existing guarantees focus on the simple case of a one-dimensional (scalar) hyperparameter. This leaves the practically important, multi-dimensional hyperparameter tuning setting unresolved. We address this open question by establishing the first general framework for establishing generalization guarantees for tuning multi-dimensional hyperparameters in data-driven settings. Our approach strengthens the generalization guarantee framework for semi-algebraic function classes by exploiting tools from real algebraic geometry, yielding sharper, more broadly applicable guarantees. For completeness, we also instantiate the first lower bound for this general setting. We further extend the analysis to hyperparameter tuning using the validation loss under minimal assumptions, and derive improved bounds when additional structure is available. Finally, we demonstrate the scope of the framework with new learnability results, including data-driven weighted group lasso and weighted fused lasso.

This paper develops a theory of scientific and technological peer effects to study how individuals' productivity responds to the behavior and network positions of their collaborators across both scientific and inventive activities. Building on a simultaneous equation network framework, the model predicts that productivity in each activity increases in a variation of the Katz-Bonacich centrality that captures within-activity and cross-activity strategic complementarities. To test these predictions, we assemble the universe of cancer-related publications and patents and construct coauthorship and coinventorship networks that jointly map the collaboration structure of researchers active in both spheres. Using an instrumental-variables approach based on predicted link formation from exogenous dyadic characteristics, and incorporating community fixed effects to address endogenous network formation, we show that both authors' and inventors' outputs rise with their network centrality, consistent with the theory. Moreover, scientific productivity significantly enhances technological productivity, while technological output does not exert a detectable reciprocal effect on scientific production, highlighting an asymmetric linkage aligned with a science-driven model of innovation. These findings provide the first empirical evidence on the joint dynamics of scientific and inventive peer effects, underscore the micro-foundations of the co-evolution of science and technology, and reveal how collaboration structures can be leveraged to design policies that enhance collective knowledge creation and downstream innovation.

Standard count models such as the Poisson and Negative Binomial models often fail to capture the large proportion of zero claims commonly observed in insurance data. To address such issue of excessive zeros, zero-inflated and hurdle models introduce additional parameters that explicitly account for excess zeros, thereby improving the joint representation of zero and positive claim outcomes. These models have further been extended with random effects to accommodate longitudinal dependence and unobserved heterogeneity. However, their consistency with fundamental probabilistic principles in insurance, particularly stochastic monotonicity, has not been formally examined. This paper provides a rigorous analysis showing that standard counting random-effect models for excessive zeros may violate this property, leading to inconsistencies in posterior credibility. We then propose new classes of counting random-effect models that both accommodate excessive zeros and ensure stochastic monotonicity, thereby providing fair and theoretically coherent credibility adjustments as claim histories evolve.

Estimating causal effects from longitudinal trajectories is central to understanding the progression of complex conditions and optimizing clinical decision-making, such as comorbidities and long COVID recovery. We introduce C-kNN–LSH, a nearest-neighbor framework for sequential causal inference designed to handle such high-dimensional, confounded situations. By utilizing locality-sensitive hashing, we efficiently identify “clinical twins” with similar covariate histories, enabling local estimation of conditional treatment effects across evolving disease states. To mitigate bias from irregular sampling and shifting patient recovery profiles, we integrate neighborhood estimator with a doubly-robust correction. Theoretical analysis guarantees our estimator is consistent and second-order robust to nuisance error. Evaluated on a real-world Long COVID cohort with 13,511 participants, C-kNN-LSH demonstrates superior performance in capturing recovery heterogeneity and estimating policy values compared to existing baselines.

We introduce a transfer learning framework for regression that leverages heterogeneous source domains to improve predictive performance in a data-scarce target domain. Our approach learns a conditional generative model separately for each source domain and calibrates the generated responses to the target domain via conditional quantile matching. This distributional alignment step corrects general discrepancies between source and target domains without imposing restrictive assumptions such as covariate or label shift. The resulting framework provides a principled and flexible approach to high-quality data augmentation for downstream learning tasks in the target domain. From a theoretical perspective, we show that an empirical risk minimizer (ERM) trained on the augmented dataset achieves a tighter excess risk bound than the target-only ERM under mild conditions. In particular, we establish new convergence rates for the quantile matching estimator that governs the transfer bias-variance tradeoff. From a practical perspective, extensive simulations and real data applications demonstrate that the proposed method consistently improves prediction accuracy over target-only learning and competing transfer learning methods.

We develop a framework for edge-wise inference in graphon-based random networks. The proposed leave-one-out neighborhood procedure excludes the target vertex when selecting neighbors, then averages the selected vertices' edges to that target. Conditional on the latent probabilities and selected neighborhood, these edges are independent. This structure yields finite-sample concentration inequalities and a Berry–Esseen bound for fluctuations about the selected-neighborhood mean. Under blockwise graphon regularity, quantitative two-hop identifiability, and across-block separation, we control the smoothing bias uniformly over latent positions, including positions near block boundaries. Together with an observable variance estimator, this gives Gaussian confidence intervals for individual edge probabilities with asymptotically correct pointwise coverage. We also establish finite-sample empirical Bernstein coverage for the selected-neighborhood mean, Gaussian inference under weaker bandwidth conditions on identical-profile blocks, and inference for target pairs chosen measurably from the latent positions. Finally, we derive an expected-risk identity for leave-one-out cross-validation at each fixed neighborhood size. Simulations assess interval coverage, variance estimation, and estimation error.

Testing whether two multivariate samples exhibit the same extremal behavior is an important problem in various fields including environmental and climate sciences. While several ad-hoc approaches exist in the literature, they often lack theoretical justification and statistical guarantees. On the other hand, extreme value theory provides the theoretical foundation for constructing asymptotically justified tests. We combine this theory with Kullback-Leibler divergence, a fundamental concept in information theory and statistics, to propose a test for equality of extremal dependence structures in practically relevant directions. Under suitable assumptions, we derive the limiting distributions of the proposed statistic under null and alternative hypotheses. Importantly, our test is fast to compute and easy to interpret by practitioners, making it attractive in applications. Simulations provide evidence of the power of our test. In a case study, we apply our method to show the strong impact of seasons on the strength of dependence between different aggregation periods (daily versus hourly) of heavy rainfall in France.

We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our en-to-end formal infrastructure implement the missing contents in latest Lean library, including a complete development of Gaussian Lipschitz concentration, Dudley's entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is provided in https://github.com/YuanheZ/lean-stat-learning-theory.

We study reinforcement learning for revenue management with delayed feedback, where a substantial fraction of value is determined by customer cancellations and modifications observed days after booking. We propose choice-model-assisted RL: a calibrated discrete choice model is used as a fixed partial world model to impute the delayed component of the learning target at decision time. In the fixed-model deployment regime, we prove that tabular Q-learning with model-imputed targets converges to an $O(\varepsilon/(1-\gamma))$ neighborhood of the optimal Q-function, where $\varepsilon$ summarizes partial-model error, with an additional $O(t^{-1/2})$ sampling term. Experiments in a simulator calibrated from 61{,}619 hotel bookings (1{,}088 independent runs) show: (i) no statistically detectable difference from a maturity-buffer DQN baseline in stationary settings; (ii) positive effects under in-family parameter shifts, with significant gains in 5 of 10 shift scenarios after Holm–Bonferroni correction (up to 12.4%); and (iii) consistent degradation under structural misspecification, where the choice model assumptions are violated (1.4–2.6% lower revenue). These results characterize when partial behavioral models improve robustness under shift and when they introduce harmful bias.

Spatial epidemiology identifies the drivers of elevated population-level disease risks, using disease counts, exposures and known confounders at the areal unit level. Poisson regression models are typically used for inference, which incorporate a linear/additive regression component and allow for unmeasured confounding via a set of spatially autocorrelated random effects. This approach requires the confounder interactions and their functional relationships with disease risk to be specified in advance, rather than being learned from the data. Therefore, this paper proposes the SPAR-Forest-ERF algorithm, which is the first fusion of random forests for capturing non-linear and interacting confounder-response effects with Bayesian spatial autocorrelation models that can estimate interpretable exposure response functions (ERF) with full uncertainty quantification. Methodologically, we extend existing methods set in a prediction context by propagating uncertainty between both the ML and statistical models, developing a new stopping criteria designed to ensure the stability of the primary inferential target, and incorporating a range of different ERFs for maximum model flexibility. This methodology is motivated by a new study quantifying the impact of air pollution concentrations on self-rated health in Scotland, using data from the recently released 2022 national census.

Instrumental variable (IV) methods are widely used to infer treatment effects in the presence of unmeasured confounding. In this paper, we study nonparametric inference with an IV under a separable binary treatment choice model, which posits that the odds of the probability of taking the treatment, conditional on the instrument and the treatment-free potential outcome, factor into separable components for each variable. While nonparametric identification of smooth functionals of the treatment-free potential outcome among the treated, such as the average treatment effect on the treated, has been established under this model, corresponding nonparametric efficient estimation has proven elusive due to variationally dependent nuisance parameters defined in terms of counterfactual quantities. To address this challenge, we introduce a new variationally independent parameterization based on nuisance functions defined directly from the observed data. This parameterization, coupled with a novel fixed-point argument, enables the use of modern machine learning methods for nuisance function estimation. We characterize the semiparametric efficiency bound for any smooth functional of the treatment-free potential outcome among the treated and construct a corresponding semiparametric efficient estimator without imposing any unnecessary restriction on nuisance functions. Furthermore, we describe a straightforward generative model justifying our identifying assumptions and characterize empirically falsifiable implications of the framework to evaluate our assumptions in practical settings. Our approach seamlessly extends to nonlinear treatment effects, population-level effects, and nonignorable missing data settings. We illustrate our methods through simulation studies and an application to the Job Corps study.

We develop a semiparametric continuous time inferential framework for testing cumulative treatment effects, including both immediate and carryover effects. The latent state process is modeled by an ordinary differential equation (ODE) whose drift depends on state and treatment, while the observed data are noisy measurements of this latent state at irregular, multi resolution time points. We define the average treatment effect (ATE) as the difference of expected cumulative outcomes under the always treat and always control policies and establish its identifiability from observational data. More generally, the framework accommodates contrasts between arbitrary deterministic stationary treatment policies, including clinically motivated threshold policies, with the always treat versus never treat comparison as the leading special case. A two step estimation procedure first recovers the latent state trajectories and their derivatives via spline smoothing, then solves a continuous time estimating equation derived from the infinitesimal generator of the ODE. We prove asymptotic normality of the resulting test statistic under explicit rate conditions on the spline estimator. The resulting test admits asymptotically valid inference under noisy, irregularly sampled observations, providing a studentized statistic with a standard $N(0,1)$ limit. Simulations show that the proposed method demonstrates improved stability and power over discrete time alternatives in noisy and multi resolution settings, with comparable performance under clean regular grids. We apply the method to the OhioT1DM diabetes dataset to evaluate cumulative effects of insulin treatments on glucose levels.

In biomedical research, repeated measurements within each subject are often processed to remove artifacts and unwanted sources of variation. The resulting data are used to construct derived outcomes that act as proxies for scientific outcomes that are not directly observable. Although intra-subject processing is widely used, its impact on inter-subject statistical inference has not been systematically studied, and a principled framework for causal analysis in this setting is lacking. In this article, we propose a semiparametric framework for causal inference with derived outcomes obtained after intra-subject processing. This framework applies to settings with a modular structure, where intra-subject analyses are conducted independently across subjects and are followed by inter-subject analyses based on parameters from the intra-subject stage. We develop multiply robust estimators of causal parameters under rate conditions on both intra-subject and inter-subject models, which allows the use of flexible machine learning. We specialize the framework to a mediation setting and focus on the natural direct effect. For high dimensional inference, we employ a step-down procedure that controls the exceedance rate of the false discovery proportion. Simulation studies demonstrate the superior performance of the proposed approach. We apply our method to estimate the impact of stimulant medication on brain connectivity in children with autism spectrum disorder.

Neural networks represent more features than they have dimensions via superposition, forcing features to share representational space. Current methods decompose activations into sparse linear features but discard geometric structure. We develop a theory for studying the geometric structre of features by analyzing the spectra (eigenvalues, eigenspaces, etc.) of weight derived matrices. In particular, we introduce the frame operator $F = WW^\top$, which gives us a spectral measure that describes how each feature allocates norm across eigenspaces. While previous tools could describe the pairwise interactions between features, spectral methods capture the global geometry (“how do all features interact?”). In toy models of superposition, we use this theory to prove that capacity saturation forces spectral localization: features collapse onto single eigenspaces, organize into tight frames, and admit discrete classification via association schemes, classifying all geometries from prior work (simplices, polygons, antiprisms). The spectral measure formalism applies to arbitrary weight matrices, enabling diagnosis of feature localization beyond toy settings. These results point toward a broader program: applying operator theory to interpretability.

In this paper, we establish Berry–Esseen bounds for both self-normalized and non-self-normalized sums of locally dependent random variables. The proofs are based on Stein's method together with a concentration inequality approach. We develop a new class of concentration inequalities that extend classical results and achieve optimal convergence rates under more general dependence structures. As applications, we apply our main results to derive sharper Berry–Esseen bounds for graph dependency, distributed $U$-statistics, constrained $U$-statistics, and decorated injective homomorphism sums.

We propose a general method to carry out a valid Bayesian analysis of a finite-dimensional `targeted' parameter in the presence of a finite-dimensional nuisance parameter. We apply our methods to causal inference based on estimating equations. While much of the literature in Bayesian causal inference has relied on the conventional 'likelihood times prior' framework, a recently proposed method, the 'Linked Bayesian Bootstrap', deviated from this classical setting to obtain valid Bayesian inference using the Dirichlet process and the Bayesian bootstrap. These methods rely on an adjustment based on the propensity score and explain how to handle the uncertainty concerning it when studying the posterior distribution of a treatment effect. We examine theoretically the asymptotic properties of the posterior distribution obtained and show that our proposed method, a generalized version of the 'Linked Bayesian Bootstrap', enjoys desirable frequentist properties. In addition, we show that the credible intervals have asymptotically the correct coverage properties. We discuss the applications of our method to mis-specified and singly-robust models in causal inference.

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\alpha>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.