Papers

This paper introduces a novel regression framework for modeling response variables restricted to the unit interval by proposing unit-Lindley additive partially linear models (UL-APLMs). This model class combines parsimony and interpretability of one-parameter unit-Lindley distribution with the flexibility of additive partial linear structures, enabling the coexistence of linear and smooth covariate effects. Additive terms are modeled using B-spline basis under a penalized likelihood framework to ensure smoothness. Estimation is carried out by maximizing the penalized log-likelihood function. The goodness-of-fit of the models is assessed through residual analysis, whereas the robustness of the parameter estimates and the detection of influential data points are evaluated using the local influence approach, which incorporates curvature diagnostics under case-weight and response perturbation schemes. The sensitivity and penalized observed information matrices are derived explicitly for the proposed model. Simulation studies demonstrate the accuracy of the estimation procedure under various scenarios. Real data on the assessment of the psychological profile of patients with hypopituitarism illustrate the applicability of the model, highlighting the diagnostic importance.

Subgroup analyses are central to the assessment of benefits and risks, where recommendations may depend on evidence that treatment effects differ across patient groups. Valid subgroup claims require evidence based on (within-trial) interaction estimates while accounting for the heterogeneity in those interaction effects. In the common case of only a few available studies, inference may benefit from the use of prior information on the expected amount of heterogeneity. Although between-study heterogeneity~($\tau$) has been studied empirically for overall treatment effects, no such calibration exists for treatment-by-subgroup interaction effects. We derive empirical (predictive) prior distributions for overall and interaction effect heterogeneity from over 3{,}000 interaction meta-analyses drawn from the Cochrane Database of Systematic Reviews (CDSR). The resulting effect-measure-specific priors indicate that interaction heterogeneity tends to be substantially smaller than treatment effect heterogeneity. We also show that lower precision of within-trial interaction estimates makes interaction heterogeneity harder to identify. Therefore, the use of empirical priors is particularly valuable in sparse interaction meta-analyses. A motivating example illustrates how priors tailored to interaction effects may substantially improve precision in a meta-analysis compared with standard heterogeneity priors.

We study sequential testing and Bayesian quickest detection for diffusion observations whose drift changes between two alternatives while the signal-to-noise ratio may depend on the current observation. In this setting the posterior probability is generally not a closed one-dimensional Markov statistic: the natural sufficient state is the augmented process consisting of the posterior (or likelihood ratio) and the observed diffusion. We formulate both testing and quickest detection within this common filtering framework and identify the corresponding degenerate free-boundary problems. The main contribution is a delay-penalty comparison principle. For a common terminal false-alarm or terminal decision cost, a pointwise larger running delay penalty increases the value of continuation, shrinks the continuation region, and yields earlier stopping. When the stopping set has a one-sided posterior representation, this gives an order relation for the optimal alarm boundaries. The result applies to linear delay costs and to nonlinear marginal delay penalties after the appropriate Markovian augmentation, and is illustrated by a constant signal-to-noise Shiryaev example in which the alarm threshold is computed numerically and shown to be monotone in the delay cost. The framework clarifies how state-dependent information and nonlinear delay costs jointly affect the geometry of sequential testing and quickest-detection rules.

State–space models provide a flexible framework for analyzing dynamical systems, yet they often rely on Gaussian assumptions that fail to capture heavy-tailed or outlier-prone measurement noise. We propose a robust estimation scheme for linear state–space models subject to compound-Gaussian noise, as encountered for instance in radio interferometry affected by radio-frequency interference (RFI). The method relies on a Stochastic Approximation Expectation–Maximization (SAEM) algorithm in which the standard E-step is replaced by Monte Carlo sampling of the latent states and noise texture through closed-form Gibbs updates, enabling tractable inference despite the heavy-tailed likelihood. Numerical experiments show that the proposed method significantly improves reconstruction fidelity and robustness to RFI, outperforming a Gaussian EM algorithm and even an oracle RTS smoother. These results highlight the benefits of heavy-tailed state–space modeling and SAEM-based inference in interference-dominated imaging scenarios.

In this paper we propose estimators of the parameter functions in an Aalen additive hasard regression model. The estimators are the individual and componentwise $l^2$ projections of the naive estimators resulting from the ordinary least squares estimator in the Aalen additive hazard model on the space of monotone functions. We provide pointwise limit distribution results for the resulting estimators, that exhibit $n^{-1/3}$ rate of convergence and the Chernoff distribution as the limit distribution.

From climate teleconnections to gene regulation, modern time-series datasets encompass tens or hundreds of interacting variables, making causal discovery increasingly challenging. Constraint-based methods offer statistical rigor but their nonlinear CI tests are infeasible at scale, while score-based alternatives avoid CI testing but require arbitrary thresholds to binarize continuous edge scores. We propose GRACE ($\textbf{G}$ated $\textbf{R}$efinement for $\textbf{A}$ccurate $\textbf{C}$ausal $\textbf{E}$dge discovery), which refines constraint-based discovery using Hard Concrete gates with $L_0$ regularization: each candidate edge has an independent gate whose values concentrate near 0 or 1, yielding a clean bimodal separation that makes the binary decision robust, unlike the narrow, overlapping score distributions produced by $L_1$ and attention-based methods. A fast linear CI skeleton provides high-recall candidates; a single gated model then prunes false positives by learning which edges genuinely improve prediction, with automatic regularization adapted to problem dimensions and skeleton density. Systematic experiments on synthetic benchmarks, spanning diverse graph topologies (scale-free, Erdős-R'enyi, small-world) and dimensionalities up to $d=100$, show that GRACE substantially improves F1 over its base CI method while maintaining high precision, and outperforms attention-based and score-based alternatives. GRACE matches or exceeds expensive nonlinear CI tests at a fraction of the cost ($75\times$ faster). On a real-world river flow dataset, where rainfall confounders, variable propagation lags, and distributional shifts violate standard assumptions, a temporal bootstrap variant of GRACE recovers 9 of 11 causal edges along the Elbe River with only 1 false positive ($F_1 = 0.86$, AUROC${} = 0.99$), reducing the skeleton's 106 false positives by 99%.

Survival analysis is central to clinical decision-making, yet reliable time-to-event models require large, diverse cohorts that are rarely available at a single institution, while privacy regulations restrict the centralization of patient data. Federated learning (FL) offers a privacy-preserving alternative by training shared models without exchanging raw data, but its effectiveness for survival modeling under realistic, heterogeneous conditions remains insufficiently understood. This paper presents a systematic, multi-model evaluation of federated survival analysis on a cross-institutional breast cancer cohort with naturally heterogeneous distributed clients. Three representative survival models, the Cox Proportional Hazards model, DeepSurv, and Random Survival Forest (RSF), are compared across centralized, local, and federated training, and three federated optimization strategies (FedAvg, FedProx, and FedAdam) are assessed for the gradient-based models. Results show that FL consistently outperforms local training and approaches, and occasionally exceeds, centralized performance, while RSF offers the best overall balance of discrimination, calibration, and robustness across heterogeneous clients. We further find that performance depends on the diversity of client distributions, and that FedAvg and FedProx are stronger and more stable than FedAdam. Based on these findings, we derive practical, decision-oriented guidelines mapping data, privacy, interpretability, and resource constraints to recommended model and training-paradigm choices for federated survival modeling in healthcare.

The exact computation of the Normalized Maximum Likelihood (NML) codelength for regular non-smooth estimators (e.g., Lasso) has been historically limited by the cubic scaling walls of manifold-constrained projection and volume integration. At each step of the geometric Propose-and-Project Metropolis–Hastings (PPMH) sampler, evaluating the projection operator requires inverting an $(N+k) \times (N+k)$ generalized KKT matrix, while calculating the volume factor requires the determinant of an $(N-k) \times (N-k)$ Gram matrix. This paper presents an exact, mathematically equivalent formulation that bypasses both bottlenecks by utilizing the block Schur complement and Sylvester's determinant identity. We prove that the computational complexity of both operations collapses from $\mathcal{O}(N^3)$ to $\mathcal{O}(k^3 + N^2 k)$ per step. We generalize this reduction to Sparse Support Vector Machines (SVMs), Elastic Net, and Group Lasso. Finally, we provide a rigorous numerical stability analysis and evaluate the sampler's efficiency using the Effective Sample Size (ESS) per second. Our empirical benchmarks on high-dimensional datasets confirm a constant speedup exceeding $14{,}100\times$ while maintaining double-precision numerical equivalence, rendering exact non-smooth NML estimation highly tractable for large-scale statistical inference.

Synthetic control (SC) methods are among the most widely used tools for causal inference without randomization. The standard Gaussian confidence interval around the estimated effect is simple, fast, and reliably directional when the treatment signal is strong, so practitioners default to it for good reason. Most treated populations, however, are bottom-heavy in intensity, and for them the SC model's systematic bias rivals or exceeds the signal even under good pre-treatment fit. Because this bias shares sign across units it does not average out, and the Gaussian confidence interval shrinks past it and converges on a wrong center. The failure is not imprecision but misdirection: a positive effect estimated as negligible is a missed opportunity, while a negligible effect estimated as significantly positive leads to continued investment in an intervention that is not working. No existing confidence interval for the SC effect measures this bias. We propose a placebo-in-time bootstrap that estimates the bias distribution directly from the observed panel. For each treated unit the procedure backdates the treatment onset and refits the SC model at each placebo onset; the resulting placebo gaps are draws from the same bias distribution that contaminates the real estimate, and bootstrapping them yields a critical value calibrated at the zero null. Because the method resamples realized model error rather than a hypothesized effect, coverage is trajectory-agnostic: it holds at fixed width regardless of how the true effect evolves over time.

In enterprise software, a contract commits the customer to a usage volume over a fixed term in exchange for discounted pricing. These contracts are individually negotiated across many dimensions – size, duration, industry, product mix, usage history – and without a data-driven reference point, discounts tend to be overly generous. Manual governance review enforces discipline but at days-scale per contract, with inconsistency across reviewers and no real-time feedback to sellers. We present Contract Scoring, a peer-based grading system deployed on every contract at Databricks. The system identifies empirically similar historical contracts via adaptive nearest neighbors over ensemble trees, where shared leaf membership defines a data-driven similarity learned from the discount target. It returns a letter grade with per-product-line breakdown in seconds; the underlying peer set is available to the centralized review team for audit. Sellers treat the grade as a contract design “exit criterion”, iteratively adjusting discount structures until the grade reflects their intended tradeoff. Deployment evidence shows measurable discount discipline across the scored portfolio, with a commercially significant impact on revenue.

Flexible continuous univariate distributions with bounded support are essential for accurate input modeling in stochastic simulation and decision analysis. Although Bézier distributions provide a powerful family capable of representing complex shapes, their adoption has been hindered by the lack of efficient fitting procedures and modern software implementations. This paper develops a computational framework for fitting Bézier distributions to empirical data via both minimum error and maximum likelihood estimation, leveraging first-order optimization methods and exploiting the geometry of the parameter space. We identify provably (asymptotically) lossless convex restrictions of the feasible set that enable efficient projection operators based on isotonic regression and develop first-order algorithms that reduce computational runtime by three to four orders of magnitude compared to traditional derivative-free methods, while delivering consistent fits across real-world data. When benchmarked against the nonlinear solver IPOPT, our methods prove three orders of magnitude faster on average and more robust, while achieving comparable accuracy. To bridge the gap between theory and practice, we introduce bezierv, an open-source Python package providing a unified interface for fitting, analyzing, and convolving Bézier distributions.

When two or more parameters or labels produce similar data, they are degenerate, or hard to distinguish. Degeneracies render both label prediction and inverse problems difficult, since both machine learning algorithms and probabilistic samplers rely on the distinguishability of data and its gradients with respect to parameters. However, identifying degeneracies in physical models or real-world datasets can be elucidating about the choice of model or the underlying process that produces the data. We present the degeneracy distillery, a method that (1) detects and (2) resolves degenerate parameter combinations (a) automatically and (b) symbolically, from parameter-data (or parameter-simulation) pairs alone, through estimation and flattening of the Fisher information matrix. By exploring the information geometry of the likelihood, we characterize degeneracies as an intrinsic property of the physical model, requiring no realised data observation. We demonstrate our approach on a range of synthetic and real-world problems, discovering symbolic coordinate transformations that identify the combinations of parameters of a model which yield independent effects on the data. The resulting coordinates flatten the Fisher information in expectation globally, in contrast to posterior-based methods that flatten only at a single point, and substantially reduce the simulation budget required for downstream neural posterior estimation. In test cases we require up to $10\times$ fewer simulations for posterior estimation at matched validation calibration whilst simultaneously gaining physical insight on the system.

Quantum tomography and quantum retrodiction are traditionally viewed as separate inference tasks: tomography reconstructs quantum states from measurement data, whereas retrodiction infers past quantum states from observed outcomes. We show that the two are manifestations of the same underlying principle. We prove that the Petz recovery map associated with a measurement channel is precisely the gradient update of the log-likelihood used in maximum-likelihood tomography. Consequently, repeated applications of the Petz map monotonically increase the likelihood. Extending beyond measurement channels, we derive a noncommutative generalization of the Petz map from the gradient of a generalized likelihood for arbitrary quantum channels. The resulting iterative procedure maximizes the likelihood and provides a general framework for quantum tomography, establishing a direct bridge between retrodiction, recovery maps, and statistical inference.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Large language models (LLMs) are increasingly woven into expert cognitive work in daily research, yet we know little about how human expertise should adapt when an AI system can execute substantial technical reasoning on its own. Here we use statistical proof development, a demanding and structured form of expert reasoning, as a window into this broader question. Drawing on day-to-day proof problems, we find that current general-purpose LLMs occupy a useful but limited role: they can execute technical components given a precisely formulated problem and targeted guidance, but become unreliable when the problem is open-ended or requires a long reasoning chain with multiple interdependent steps. This execution-strategy gap is rooted in what makes research-level statistical proof distinctive: unlike pure mathematics, where problems arrive pre-formulated and often demand novel techniques, statistical proof requires first modeling a scientific question into a statistical framework with appropriate assumptions, and then identifying and adapting the right strategy from a repertoire of reusable domain-specific tools. Each step requires deep expertise in both the statistical literature and the real-world context being modeled. In such work, current AI assistance does not reduce the need for human expertise; it relocates that expertise to where human decision-making matters most, such as problem formulation and verification of AI-generated results, and may raise the bar for both. These findings yield practical suggestions for how statisticians can structure AI-assisted proof workflows, and point to a broader community agenda for shared resources, better AI tools, and training the next generation of researchers. Using statistical proof as a window, our study has implications for how experts structure human-AI collaboration in technical cognitive domains more broadly.

Bayesian experimental design (BED) has traditionally been based on maximising expected uncertainty reductions from prior to posterior. A major shortfall of this approach is that it leads to doubly intractable objectives that are difficult to optimise, while customising them to particular downstream tasks of interest can also be difficult. Following first principles decision theory, we demonstrate that BED can alternatively be formulated in terms of an expected future loss (EFL) on downstream actions, providing a simple and naturally task-driven framework. Critically, we then show that all such EFLs can be rearranged into singly intractable objectives that can be jointly optimised with respect to both the design policy and a downstream action policy using stochastic gradients, an approach we refer to as ACTION-BED. This formulation further sidesteps the need for any explicit posterior or marginal likelihood estimation and is naturally implicit, requiring only the ability to sample from the joint model over model parameters and data, and evaluate the downstream loss function. It thus allows design policies to be learned more effectively, efficiently, and simply than existing methods, while providing easy customisation to different downstream tasks and losses.

Exposure misclassification is a common issue prevalent in studies of respiratory infections in cystic fibrosis. Throat swabs are frequently used in place of expectorated or induced sputum cultures, although they have imperfect sensitivity and specificity to detect Pseudomonas aeruginosa and Staphylococcus aureus. We develop calibration weighting and control variate estimators for causal inference with multiple misclassified binary exposures and clustered observations. The calibration approach treats misclassification as a missing data problem, achieving consistency without modeling the misclassification mechanism. The control variate adjustment integrates information from error-prone observations to reduce variance while preserving the consistency of the gold-standard estimator. We show that the resulting estimator inherits double robustness from its component estimators. We also characterize a structural ceiling on efficiency gains in the bivariate setting, where joint correct classification of both exposures limits the variance reduction achievable relative to univariate applications. Simulation studies confirm the consistency and double robustness of the proposed estimators under model misspecification. We then apply these methods to a cohort of $651$ cystic fibrosis patients ages $6$–$21$. Swab-based estimates attenuate the effect of P. aeruginosa on percent predicted FEV$_1$ by approximately $79\%$ relative to sputum-based estimates ($-1.68$ vs. $-7.88$ percentage points; $95\%$ confidence interval for sputum: $-10.08$, $-5.67$). These findings suggest that relying on throat swabs may lead to under-treatment of P. aeruginosa infections. More broadly, the methods provide a framework for causal inference with multiple misclassified exposures.

We present an efficient protocol for learning an unknown $k$-local Lindblad generator on $n$ qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed $k$ and a supplied weighted interaction-strength bound $\bar\alpha$, the protocol estimates all Hamiltonian and dissipative Pauli–GKSL coefficients with total coefficient error at each site at most $\varepsilon$ with probability at least $1-\delta$ using $\widetilde{\mathcal O}_k(\bar\alpha^2n^{2k-2}\varepsilon^{-2}\log(n/\delta))$ samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid $k$-local Lindblad generator with diamond-norm error at most $\varepsilon$ using $\widetilde{\mathcal O}_k(\bar\alpha^2n^{2k}\varepsilon^{-2}\log(n/\delta))$ samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound $r$ and a sufficiently small sitewise subthreshold tail $\mathcal O_k(\varepsilon)$, coefficient recovery requires only $\widetilde{\mathcal O}_k(\bar\alpha^2r^2\varepsilon^{-2}\log(n/\delta))$ samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on $n$. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.

In many modern applications of reinforcement learning (RL), the natural reward for a task of interest is inherently sparse: a reward of 0 is given everywhere except when the task is completed, when a reward of +1 is given. Training a policy to maximize such a sparse reward requires solving a challenging credit assignment problem, leading to slow or ineffective RL improvement. We propose a simple approach to transform a sparse outcome reward into a dense process reward. Our approach relies on training a discriminator to distinguish between previous successful and unsuccessful episodes, and using this discriminator to incentivize the RL-learned policy to match the state-action visitations of successful episodes, while avoiding those of unsuccessful episodes. By incentivizing the policy to match the visitations over all states, not just those that correspond to task success, this reward provides dense feedback on whether progress is being made towards task completion, and, we show, provably achieves this without changing the optimal policy. Focusing on finetuning of robotic control policies, we demonstrate that our approach leads to significantly faster RL finetuning performance on both simulated and real-world manipulation tasks, as compared to simply maximizing the sparse outcome reward.

Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that $\widetilde{O}(k/\varepsilon)$ iterations suffice to generate an $\varepsilon$-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

Recent attempts to combine large language models (LLMs) with causal discovery ask models to infer pairwise directions, propose graph structures, or inject language-model outputs as priors and constraints. These approaches promise faster analysis, but they also obscure whether a causal evidence is supported by data and assumptions or by textual associations, prompt artifacts and hallucinated mechanisms. We argue for a different role for agents in causal discovery. Agents should inspect data, retrieve context, explain method assumptions and clarify graph outputs, but they should not supply edges, orientations, priors, constraints or causal conclusions. We propose the principle that agents assist the workflow, while causal claims remain grounded in data, explicit assumptions, formal algorithms, diagnostics and user or domain-expert decisions. We instantiate this principle in causal-learn+, an online platform that coordinates data analysis, preprocessing, method recommendation, expert-knowledge incorporation, formal discovery and interpretation around the algorithmic ecosystem of causal-learn. A case study on Big Five personality data illustrates agent-assisted pipeline of causal discovery without turning language-model unreliability into causal evidence. The platform is available at causallearn.com.

Neural networks are a commonly used prediction tool in computer science and statistics. However, the barrier to entry of this interesting field remains high, particularly for classical statisticians trained in a frequentist perspective. In this letter, we demystify neural networks by describing networks that approximate a linear regression and describe common customizations that provide a foundation for further study.

Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

We develop an operator-theoretic framework for finite-dimensional, regime-dependent Volterra equations with completely monotone memory kernels, dissipative network coupling, and Hawkes-type self-excitation. For each fixed regime we construct the associated Volterra resolvent family and prove global well-posedness, continuity across regime switches, and explicit a priori bounds. The main stability result is sharp in the commuting case: after simultaneous diagonalization of the network Laplacian and the excitation operator, each mode obeys a scalar characteristic equation, and global asymptotic stability holds exactly when every modal branching ratio lies below the intensity damping threshold. We also give a norm-based sufficient condition for noncommuting operators and a Perron–Frobenius spectral criterion for nonnegative intensity blocks, showing when norm estimates are conservative. Beyond mean stability, we prove a pathwise finite-range power law for burst amplitudes generated by residence in a Hurwitz but nonnormal regime: under a cone-alignment event, the survival exponent is the ratio of the regime exit rate to a cone-corrected finite-time growth rate bounded above by the logarithmic norm of a fixed Markovian realization in the chosen Euclidean metric. A complementary idealized-feedback result shows how a logarithmic-norm contraction caps the amplification band. Finally, we derive the deterministic intensity block as a mean-field limit of a relaxing long-memory Hawkes system with regimes. Numerical experiments on modal equations, a small-world network, and a switched nonnormal ODE validate the sharp threshold and the finite-range amplification mechanism without using the closed-form tail formula as input.

Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose the topology into separate ranks, leaving practitioners to fuse the information back to vertices through ad hoc choices. We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing, and enables the inclusion of topological features in neural network architectures via positional encoding. The project page can be found http://circle-group.github.io/research/CollapsedEffectiveOperators

Bayesian experimental design (BED) has traditionally focused exclusively on collecting data that is maximally informative, with little consideration of other desirable dataset characteristics, such as fairness. We therefore extend BED to target data that is both informative and fair. To this end, we introduce FairBED, which provides a formalism for the fairness of datasets themselves by measuring how compatible they are with a fair world, in which the target quantity we wish to learn about is unrelated to the sensitive attributes we want to be fair to. We then use this to construct practical fairness-aware BED objectives that maximise expected information gain about the target variables, while penalising information about sensitive attributes and/or their coupling with the target. We further extend FairBED to active learning, and show empirically that models trained on data gathered using FairBED provide improved fairness-accuracy trade-offs compared to randomly acquired data and conventional BED.

We study how the choice of default prior for a common Gaussian scale affects high-dimensional shrinkage risk, highlighting the role played by high-dimensional geometry. Formally, we consider a high-dimensional setting in which the near-zero behavior of the common scale prior has first-order consequences for shrinkage risk, and show that priors that are flat on the variance and those flat on the standard deviation allocate markedly different mass near the zero-scale boundary, leading to distinct shrinkage behavior and informing principled default prior selection. Specifically, under a radial-power benchmark, we establish that the SD-flat benchmark has a one-unit asymptotic risk advantage near the origin, crosses over in the critical regime, and is second-order equivalent to the variance-flat benchmark for strong signals. Proper single global-scale hyperpriors and bounded coordinate-multiplier mixtures inherit these limits through the near-zero exponent of their SD-scale density. For heavier-tailed or sparse priors, that exponent still classifies the common global-scale component, while local-scale tails, model-size priors, or allocation priors can also affect risk.

Existing power and design methods for multiple-intervention stepped wedge designs (M-SWDs) typically assume exchangeable cluster-period correlation, despite evidence that correlation often decays over time. Misspecification of this correlation structure can substantially distort variance estimation and power, particularly for treatment interaction effects. We develop a unified covariance framework for M-SWDs that separates intracluster correlation from an explicit cluster-period correlation matrix. This formulation accommodates exchangeable, autoregressive, and more general distance-dependent correlation structures while preserving closed-form expressions for the variance of treatment effect estimators under linear mixed models. Using analytic results and simulation studies, we demonstrate that assuming uniform correlation when the true structure is time-dependent can lead to substantial power mischaracterization. Specifically, we find that designs calibrated under independence assumptions may be overly conservative and compound symmetry can be either optimistic or conservative. These findings demonstrate the importance of explicitly modeling cluster-period correlation at the design stage of M-SWDs and provide practical guidance for power calculation and design selection in realistic settings.

Over the past decade, deep neural networks (DNNs) have achieved remarkable success on complex machine-learning tasks, yet the theoretical foundations of their performance remain incomplete. From a statistical viewpoint, a natural question is: can DNNs attain feature-learning and prediction consistency comparable to that of classical models? While a full characterization is open, we provide positive results for a broad subclass. We establish feature-learning consistency guarantees for sublinearly structured DNNs-architectures whose input/output dimensions and number of hidden neurons grow sublinearly with the sample size-when learning hierarchically compositional target functions. Importantly, this consistency still holds even in the conventional "over-parameterized" regime where the total number of parameters exceeds the number of training samples. Empirically, sublinearly structured DNNs match or surpass wide DNNs in prediction. A structural audit further indicates that widely used convolutional neural networks (CNNs), including AlexNet, VGGNet, ResNet, GoogLeNet, are sublinearly structured on their image classification benchmarks. We further prove that the sublinearly structured DNNs achieve universal approximation for hierarchically compositional functions in the large-sample limit. Moreover, images exhibit an inherent hierarchical, compositional structure. Taken together, these results explain, through a statistical lens, why many large-scale deep learning models succeed after adequate training on massive image datasets.

Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to discrete response variables, like class labels. This task is fundamental in domains, including health monitoring, where temporal structure is critical for prediction. Dynamic Time Warping (DTW) is a standard technique for measuring similarity between sequences varying in time or speed. However, DTW is restricted to discrete point matching. Moving beyond pairwise alignment, we propose a theoretical framework learning mappings between real-valued functions. These mappings approximate the flow associated with the characteristic curves of a linear transport equation with a space-dependent velocity field, providing a diffeomorphic transformation between time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function to learn these ODEs derives from the fundamental theorem of calculus. To enable flexible, expressive representations of the velocity field, we utilize reproducing kernel Hilbert spaces and optimal control methods. Our method, Diffeomorphic Time Warping (DiffTW), provides a theoretically grounded dissimilarity measure. Using a 1-nearest neighbor classifier, DiffTW outperforms unconstrained DTW on 39 of 85 datasets with 3 ties; however, constrained DTW outperforms DiffTW on 48 of 85 datasets with 5 ties.