Papers

Standard conformal prediction (CP) procedures are typically formulated in terms of p-values, but reliance on p-values alone limits flexibility, for example, when combining dependent evidence across models or data splits. Recent work has explored e-value formulations for conformal inference, yet a direct connection between p- and e-value formulations in CP has been missing, especially regarding their statistical efficiency. We first identify limitations of classical p-to-e calibrators in the CP setting, showing that they are not set-preserving and can lead to overly conservative prediction sets. To address this, we propose a novel P2E calibrator that converts conformal p-values into e-values without altering the prediction set induced by the original conformal p-value. We establish both theoretically and empirically that our calibrator can yield significant efficiency gains over existing p-to-e calibrators. This e-value formulation enables principled use of recent advances in e-value merging and randomization, where we demonstrate its impact in two applications: cross-conformal prediction (CCP), whose variants typically provide only approximate $1-2\alpha$ coverage, and conformal aggregation (CA). In both cases, our e-value-based methods satisfy the desired $1-\alpha$ coverage guarantee while improving efficiency over standard baselines. More broadly, our approach expands the flexibility of CP and opens new directions for efficient, distribution-free uncertainty quantification.

The discovery rate of fast radio bursts (FRBs) continues to increase with the advent of new radio facilities and yet extracting their astrophysical parameters such as scattering timescale ($\tau$) remains a significant bottleneck. Current $\tau$ measurement approaches like fitting analytic template models and scattering aware de-convolution are accurate but slow, sensitive to initialization, limited by low signal to noise and often require manual supervision. These limitations inspired us to explore fast, robust and scalable machine learning methods to estimate the astrophysical parameter value. We present a deep learning approach named Multimodal Transformer Based Generic Mixture Density Network (MT-GMDN) which ingests FRB dynamic spectrum and its corresponding timeseries profile through parallel transformer encoders, fuses their latent representations and predicts the distribution of $\tau$ with probabilistic output derived from generic mixture-density formulation. This formulation not only estimates the value of $\tau$ but also captures the (zero inflated) nature of FRB populations where a significant fraction of bursts exhibit unresolvable scattering. We trained MT-GMDN on $\sim3500$ FRBs from CHIME/FRB \cattwo while holding out some fraction of FRBs for validation during training and for testing after the training completes. The model achieves a coefficient of determination ($R^2$) value of $94\%$ on the expected value of $\tau$ for the events with measurable scattering with an excellent recall value of $90\%$ on the test data set. The model was also able to incorporate heteroskedastic errors enabling us the construction of a confidence interval for the predictions.

This work proposes a novel solution to predict pulsar timing residuals with limited data, addressing the critical challenge of data scarcity across spin-frequency subgroups of millisecond pulsars in PTA datasets. The proposed solution applies a Long Short-Term Memory (LSTM) network optimized using the model-agnostic meta-learning algorithm, enabling rapid adaptation to new frequency domain by fine-tuning the LSTM network with only a few-shot of ground truth timing residuals. Particle swarm optimization algorithm is also used for automatic hyperparameter optimization, leading to improved prediction accuracy. Our solution, evaluated on the second data release of the International Pulsar Timing Array (IPTA), demonstrates robust generalization with accurate predictions in three metrics across high-frequency test frequency domains, while requiring only 10% of the timing residuals from these domains for model fine-tuning. Furthermore, our lightweight structure only costs 16.86 MB CPU memory and 18 milliseconds for single-step residual prediction. All these characteristics make our solution highly suitable for real-world applications, where effective and real-time predictions of pulsar timing residuals are essential-particularly in resource-constrained environments with limited computational power, memory, or energy availability.

For nonconvex optimization problems whose objective is the prediction function of a trained Support Vector Regression (SVR) model with the Gaussian radial basis function (RBF) kernel (RBF-SVR), we present a framework that applies the difference of convex functions (DC) algorithm (DCA) by exploiting the analytical structure of the RBF kernel to construct an explicit DC decomposition. Specifically, we derive in closed form both the lower bound $\mu$ of the strong convexity parameter of the DC components and the upper bound $L$ of the gradient Lipschitz constant of the subproblem. Both $\mu$ and $L$ are determined solely by the post-training dual-coefficient sum $C_{\alpha}$ and the RBF kernel parameter $\gamma$, together with the DC decomposition parameter $\rho$, and they share a common leading term $C_{\alpha}\rho$. Through numerical experiments on six benchmark functions, we show that $C_{\alpha}\rho$ is the primary single quantity characterizing both the convergence properties and the initial-point dependence of DCA, and further demonstrate that it decomposes into two independent pathways, $C \to C_{\alpha}$ and $\gamma \to \rho$, with its primary variation governed by the SVR hyperparameters $(C, \gamma)$. Together, these results allow the convergence properties of DCA on RBF-SVR to be assessed in advance through the single scalar quantity $C_{\alpha}\rho$: approximately from $(C, \gamma)$ before training, and exactly in closed form after training.

While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.

In many risk prediction problems, covariates and a response surrogate are routinely available for a large target population, whereas the true response is costly to ascertain and is observed only for a limited subset. This creates a design problem: one must decide which observations should receive response measurement in order to build a prediction model under a fixed measurement budget. We propose a surrogate-assisted optimal sampling framework for risk prediction under measurement constraints. In the target setting, the surrogate identifies confirmed positive cases, while responses for surrogate-negative observations remain unobserved and can be selectively measured, and thus the sampling design determines how the response measurement budget is allocated. Our framework constructs an optimal sampling design minimizing the leading term of the expected out-of-sample cross-entropy loss and incorporates the resulting design into an inverse-probability-weighted cross-entropy estimator. The proposed design depends only on covariates, the surrogate, and a preliminary estimator, and therefore does not require responses from unlabeled observations at the design stage. We establish consistency, asymptotic normality, and leading-order prediction optimality of the resulting estimator. Extensive simulation studies and two real data applications demonstrate that the proposed design improves prediction performance and exhibits robustness under surrogate misspecification and rare outcome settings.

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

The ABC (area between curves) statistic is an $L^1$-distance which targets an easy-to-interpret estimand. Defined as the (normalized) integrated absolute distance between two survival curves it is a meaningful quantity even when survival functions are crossing. Based on right-censored time-to-event data, estimation is based on Kaplan-Meier curves obtained from two independent sample groups. In the present paper, we develop the large sample properties of the ABC statistic and investigate various resampling options for approximating the statistic's distribution which is possibly non-normal in the limit. These breakthroughs enable the construction of equivalence tests which can be used to establish that differences between two survival functions are practically irrelevant. Alternatively, the point estimator can be accompanied with confidence intervals that comprehensibly quantify the difference between the curves. An extensive simulation study explores these inferential methods under various scenarios: proportional, crossing, and partially equal survival functions. An application to data on overall and progression-free survival in a lung cancer trial illustrates the methods' benefits and some points of consideration.

Recently, the mapping of the replicator equation onto Bayes' theorem has been recognised, leading to an analogy between evolutionary dynamics and Bayesian learning. However, this analogy holds only for pure selection in infinite populations and breaks down when mutations – a central mechanism of evolution – are introduced. Here I propose that evolution by natural selection, at least for populations of haploid replicators in static environments, is best understood not as a learning process but as a process of causal inference. Each mutation event constitutes a natural experiment in which the parent serves as the control and the mutant offspring as the treated unit. Natural selection screens the causal effect of the mutation on fitness, retaining mutations with non-negative effects. I formalise this view within the Neyman-Rubin potential-outcomes framework. I first develop the general theory using a generic fitness outcome and show how the core identification assumptions in causal inference (Stable Unit Treatment Value Assumption, Consistency, Unconfoundedness, Positivity) map onto evolutionary biology. Using the unnormalised quasispecies equation, I prove that the intergenerational change in mean fitness decomposes exactly into a selection term – recovering Fisher's Fundamental Theorem – plus a mutation term that corresponds to a fitness-weighted average of the cumulated effect of all mutations over all parental genotypes. I show that this decomposition extends, under suitable assumptions, to the generalised replicator-mutator equation and that the frequencies of populations of matched parents-offspring update in proportion to the average causal effect of mutations on fitness.

Diffusion-based generative models have achieved remarkable success in high-dimensional data generation; however, they fundamentally rely on isotropic diffusion processes that destroy meaningful geometric structures in the forward process. For complex, multimodal, and highly correlated distributions such as biologically constrained genetic data, isotropic noise merges distinct modes and distorts intrinsic dependencies. This forces the reverse process to recover structure from heavily degraded signals, leading to slow convergence, mode averaging, and biologically implausible samples. To address this, we introduce the Geometry-aware Ornstein-Uhlenbeck (GOU) process, a structured drift design that embeds data geometry into forward and backward dynamics. By employing a variance-aware anisotropic drift, GOU contracts low-variance directions rapidly while preserving high-variance directions longer, maintaining key multimodal structures as stable channels over time. Crucially, we show that GOU's backward initialization error is governed by local rather than global variance. This geometry-adaptive initialization improves convergence rates by reducing initial mismatch and preserving cluster-level structures. Synthetic and real-world genetic experiments demonstrate that GOU significantly improves mode separation, correlation preservation, and statistical validity over standard isotropic models.

Score-based diffusion models have emerged as prominent deep generative models; however, their application to tabular data remains challenging because their backbones assume fully specified inputs, whereas real-world tabular data often contain missing values. We propose AugMask, a plug-and-play training framework that adapts missing-unaware backbones to incomplete data by separating conditioning from supervision. AugMask 1) constructs numeric inputs via conditional stochastic augmentation using lightweight auxiliary models, and 2) applies denoising supervision only to observed coordinates. In effect, augmented missing entries serve as uncertain conditioning context rather than training targets. We connect this training rule to a Rao–Blackwellized objective and show that marginalizing missing entries yields a variance-weighted sensitivity penalty, discouraging over-reliance on uncertain completions. Across diverse datasets and missingness regimes, AugMask enables standard diffusion-based tabular generators to outperform specialized missing-aware baselines.

Many systems used in real-world environments require adding new categories and incorporating new information without forgetting what was previously learnt by the classification model. This is known as class-incremental continual learning, and in the case of multivariate time-series, is further complicated by the temporal structure of the data. In this paper, we present a novel approach for performing class incremental continual learning for the classification of multivariate time series data based upon the construction of a dual-stream feature extraction pipeline (using both deep temporal embedding features generated via a pre-trained frozen foundation model and application of statistical features). Evaluated on five benchmark datasets, the proposed system achieves competitive average accuracy across all datasets while maintaining low forgetting rates across all experimental configurations.

In nature, events that affect some individuals or groups but not others constitute an implicit intervention and are known as natural experiments. For example, the COVID-19 pandemic was an intervention by the coronavirus on the sub-population infected with COVID. We ask, do natural experiments occur in existing real-world datasets? If yes, how should we treat them? To detect natural experiments in data, we use causal discovery to recover the underlying causal graph and perform feature selection based on causal links. If downstream performance improves by treating the data as interventional rather than observational, we argue that this suggests the dataset contains natural experiments. We first validate this hypothesis by simulating datasets with and without natural experiments using synthetic graphs. We then perform a systematic empirical evaluation on a large suite of real-world datasets. Our results indicate that real-world datasets do contain natural experiments and we can take advantage of those natural experiments to improve model performance using causal inference. Our work represents the initial foray into this area, offering a preliminary exploration within a limited scope.

Concepts of calibration formalize the compatibility between probabilistic predictions and the respective outcomes. In a nutshell, the outcomes ought to be indistinguishable from random draws from the predictive distributions. In this paper, we review, extend, and bridge notions of calibration that have been proposed for classification and regression tasks. Particular emphasis is given to hierarchical relations between the various notions, as they apply to general real-valued data, continuous outcomes, count data, nominal classes, and binary outcomes. To highlight a number of contributions, we introduce the notion of modal calibration for nominal outcomes, we distinguish full, partial, and average calibration in this setting, and we show that double probability integral transform (PIT) calibration is logically independent of previously proposed concepts of calibration for discrete outcomes. Furthermore, we generalize extant results on concepts of calibration that are expressed in terms of properties or functionals of the predictive distributions, such as means, quantiles, or event probabilities. Throughout the paper, we illustrate the concepts and their hierarchical relations in worked examples, and we provide algorithmic tools that support the construction of instructive examples and counterexamples.

This paper presents a post-Bayesian approach to online filtering in nonlinear state-space models, capable of avoiding over-confident inferences in settings where either the dynamical model, the measurement model, or both, could be misspecified. This is addressed using predictively oriented (PrO) posteriors, an emerging paradigm in which learning (i.e., posterior concentration) occurs if and only if the overall model is well-specified, without strict adherence to Bayes' theorem. As the characterisation of PrO posteriors is challenging, our main technical contribution is a fast approximate linear-Gaussian update procedure, analogous to an (iterated) extended Kalman filter. The methodology, which we call EKF-PrO, has no tunable hyper-parameters and has a computational cost comparable to that of existing filtering methods. Performance is empirically assessed on a range of linear and non-linear applications, in which the state-space model is systematically misspecified.

Chain-of-thought (CoT) reasoning has become a widely used mechanism for eliciting multi-step reasoning in large language models by generating intermediate reasoning steps at inference time. Yet the scaling behavior of generalization with CoT depth remains poorly understood. To address this question, we study a theoretically solvable model of CoT for in-context weight prediction in linear regression, where test-time reasoning is represented as an iterative refinement of the weight-parameter estimate. Using tools from random matrix theory under high-dimensional asymptotics, we derive an exact formula for the generalization error as a function of reasoning depth, pretraining data amount, and context length. Our analysis reveals a sharp phase transition separating exponential and polynomial improvement, saturation, and overthinking, and characterizes how the optimal reasoning depth scales. We further show that deeper reasoning is most effective with sufficiently rich pretraining and in-context information, whereas limited pretraining or context makes longer reasoning prone to error amplification or saturation. We also validate these predictions through experiments on fully learned linear attention and softmax attention models. Our results provide a unified theoretical account of how test-time CoT depth affects generalization.

Active Statistical Inference is a new framework to make precise claims about population parameters with provable statistical guarantees. It uses a predictive "black-box" machine learning (ML) model to strategically decide which data points to label, roughly prioritizing samples for which the ML model is unsure about their label values. A major issue is that the framework can be brittle when uncertainty estimates are noisy. This paper introduces OPAL (Optimized Policy for Allocation of Labels), which learns a labeling strategy within a tractable class of smooth policies to yield estimators with the lowest variance. In effect, OPAL is an end-to-end pipeline that turns a black-box model's uncertainty scores into a data-adaptive labeling strategy and then performs inference on the collected samples. We evaluate OPAL on real datasets spanning medical imaging data, computational social science, and proteomics. As a concrete example, we consider predicting breast cancer subtype from histopathology images and using OPAL to form valid confidence intervals for odds ratios for different demographic groups. We show that OPAL achieves nominal coverage in finite samples and has the accuracy one expects from methods which have far more labeled samples.

Tail index regression studies how covariates affect tail heaviness in heavy-tailed data. In many applications, data are distributed across heterogeneous sources, where direct pooling is infeasible due to privacy or regulatory constraints. Existing methods mainly focus on single-dataset analysis and do not address heterogeneous federated settings. We develop a personalized federated framework for high-dimensional tail index regression that accommodates client heterogeneity while exploiting latent similarities across clients. The proposed estimator combines sparsity regularization with nonconcave fusion penalties to perform coefficient estimation, variable selection, and group recovery. We establish non-asymptotic convergence rates and show that the estimator enjoys an oracle property by consistently recovering the underlying grouping structure. For computation, we develop an ADMM-based federated algorithm with adaptive gradient updates and establish its convergence guarantees. We further propose a debiased federated inference procedure based on adaptive weighted aggregation across related clients, yielding valid confidence intervals and hypothesis tests with improved efficiency over target-only inference. Simulation studies and real-data analysis demonstrate the effectiveness of the proposed methods.

With the increasing scale and number of wind farms, wind turbines' daily operation and maintenance costs are increasing. To reduce operation and maintenance costs and enhance the reliability of wind turbine and system operation data before reaching catastrophic failures, monitoring the operating status of the equipment and detecting failures at an early stage is crucial. It is of great practical significance to utilize the working condition data for abnormal assessment of the operating status of wind turbines to realize abnormal monitoring of the operating status of wind turbines. However, the existing anomaly detection methods can neither perform effective relational modeling in data filled with a large amount of redundant information nor reasonably utilize the valuable anomaly data. For this reason, this paper proposes an anomaly detection model that fuses a Transformer and a generative adversarial network. Firstly, it reduces the leakage detection rate of minor deviation anomalies by amplifying the reconstruction error. Secondly, it uses autoregressive inference to extract multimodal features to enhance the stability and generalization ability of training. Finally, the temporal feature extraction module is constructed to promote the interactive learning between features of different time scales and effectively reduce the time redundancy. The results of multiple sets of experiments conducted on real WTG datasets show that TransGAN-WT achieves an average F1 score of 96.10% across multiple wind turbine datasets, which is 5.84% and 2.89% higher than several other state-of-the-art baseline methods. It also realizes a false positive rate (FPR) of 0.06%, and is verified by the Wilcoxon signed-rank test to have achieved a statistically significant performance enhancement compared to the state-of-the-art baseline methods, effectively ensuring the stable operation of wind turbines.

A recurring data mining task in complex networks is to determine how individual nodes contribute to system behavior. Existing approaches rely on either static-graph centralities or control-theoretic quantities such as controllability Gramians, which assume linear, time-invariant dynamics. Estimated systems, however, are typically nonlinear and time-varying. We define "emergent contribution (EC)," a finite-horizon measure of a node's dynamical leverage: the metric-weighted energy of its impulse response accumulated along the system trajectory. Computed from the Jacobians of any differentiable model, EC is estimator-agnostic and reduces exactly to average controllability in the linear, time-invariant limit. Our contribution is a characterization of when the two measures agree and diverge. Using a controlled synthetic family with known ground-truth contribution, we construct a phase diagram spanning nonlinearity, regime structure, persistence, and perturbation amplitude. EC and average controllability agree under static or smoothly drifting dynamics and both track ground truth. Divergence emerges under persistent regime switching, is strongest under persistent sign reversal, and disappears when the sign reversal is removed. At extreme perturbation amplitudes, both measures degrade, identifying the limits of local linearization. We place five estimated real systems from several domains within this phase space. Their placement serves as a diagnostic of when EC provides information beyond static controllability and therefore justifies its additional computational cost. On one panel examined in depth, a twenty-seed retraining ensemble reveals a robust variance–leverage dissociation: nodes whose perturbations propagate widely despite low within-system variance, which is not recovered by static centralities nor variance-based summaries.

We study the geometry of feasible value functions in infinite-horizon partially observable Markov decision processes (POMDPs) under memoryless stochastic policies. Our main contribution is a characterization of the feasible set of value functions as a semi-algebraic set, defined by explicit polynomial inequalities determined by the transition dynamics, observation kernel, and reward structure of the POMDP. This result extends prior work for fully observable Markov decision processes, where the feasible set is known to be a polytope, to the substantially more intricate partially observable setting. In contrast to the polyhedral structure arising in MDPs, partial observability induces fundamentally nonlinear constraints, leading to a richer and more complex geometric structure. Our geometric characterization provides new insight into the landscape of policy optimization in both MDPs and POMDPs, and reveals qualitative phenomena unique to partial observability, including the emergence of isolated local maximizers of the long-term reward and their dependence on the initial state distribution.

We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the $\mathsf E_\gamma$-divergence with $\gamma\ge1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the $\mathsf E_\gamma$-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, $\chi^2$, and Rényi divergences. For independent Metropolis–Hastings with target $\pi$, proposal $q$, and unbounded importance weight $w=d\pi/dq$, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core $C_R=\{w\le R\}$ and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile $H_R=\pi(w>R)$, recovering sharp existing moment-based convergence rates when $\mathbb E_q[w^p]<\infty$ for some $p>1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p>1$ exists.

Count data with excess zeros arise frequently in health economics and epidemiology. The standard Poisson Hurdle Model (PHM) parametrises the underlying Poisson rate directly, so its count-component coefficients are log-rate ratios rather than log-ratios of the marginal mean. Consequently, the incidence density ratio (IDR) from the PHM is neither exact nor constant across covariate profiles, complicating applied reporting. We propose the Marginalised Poisson Hurdle Model (MPHM), which reparametrises the count component so that the coefficient vector beta directly governs the marginal mean E[Y]. A nonlinear connector equation links the structural Poisson rate to this parametrised mean. We prove existence and uniqueness of the connector solution, develop a vectorised Brent's-method solver, derive the score equations and block-diagonal Fisher information, establish asymptotic normality, and prove that exp(beta) is exactly constant across all covariate values. A simulation study with n in {100, 250, 500, 1000}, zero proportion pi in {0.2, 0.4, 0.6, 0.8}, and R = 200 replications confirms consistency, near-zero bias, and 95% Wald coverage of 0.905-0.975 across all 16 scenarios. Applied to the NMES1988 physician visit data (n = 4,406), the MPHM yields IDR = 1.163 (95% CI: 1.150-1.177) per additional chronic condition - an exact, population-wide effect not derivable from the PHM. The MPHM resolves the non-constant IDR problem by directly parametrising E[Y]. The resulting IDR holds for every individual and the whole population without further marginalisation, substantially simplifying the reporting of covariate effects in health utilisation research.

Modeling interactions among multimodal, high-dimensional data is intrinsically challenging due to ultra-high dimensionality and complex dependence structure with high level noise. Screening methods are effective for reducing dimensionality, but most existing approaches shrink only the predictor space while retaining all outcomes. In cross-modal analyses, different outcomes often select different predictor subsets, so the union remains large and the response dimension is unchanged, limiting the practical benefit of screening. This gives rise to heavy computational burdens and poor interpretability. To address these limitations, we propose a new screening framework, Graph Independence Dual Screening (GIDS), which simultaneously reduces the dimensionality of response variables and predictors. We design computationally efficient algorithms that facilitate downstream selection procedures, improving accuracy and scalability, and establish supporting theoretical results. Extensive simulation studies demonstrate that GIDS outperforms existing methods that screen only predictors. To illustrate its utility, we applied GIDS to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, analyzing interactions between genome-wide 865,353 DNA methylation and 49,386 transcriptomic variables. GIDS reduced the feature space to approximately 9,000 CpGs and 2,000 transcripts, uncovering blockwise interaction structures: clusters of CpG sites and gene transcripts with strong associations. These findings not only improve computational tractability but also yield interpretable biological insights, highlighting coordinated regulatory mechanisms underlying Alzheimer's disease.

Switchback experiments – in which treatment is assigned at the level of a cluster crossed with a time period – are widely used in marketplace and platform settings, yet no closed-form power formula exists for them. We fill this gap by deriving a closed-form, multi-level asymptotic variance approximation for the individual-level OLS estimator, facilitating power budgeting. Using this formula, we reveal a structural floor on statistical power: while idiosyncratic noise vanishes with observation density, macro-level shocks are multiplicatively penalized by cluster size imbalance. We confirm through analytical derivations and Monte Carlo simulations that the formula is exact across typical parameters and serves as a mathematically conservative upper bound in extreme boundary regimes. We study three methodological applications. First, we prove that advanced assignment designs like stratification only partially eliminate the penalty of cluster size imbalance on power. Second, we demonstrate that variance reduction techniques targeting macro-level shocks yield disproportionately greater efficiency gains than those targeting residual noise. Third, we formalize the finite-sample power trade-offs between individual-level and cell-level estimators.

We provide a theoretical ground to directly compute the final size distribution of a Galton–Watson process (GWP) from the implicit fixed point equations for its probability generating function (PGF). Firstly we show that valid identification of the final size PGF is ensured on any contour inside the unit polydisc. Furthermore, we derive error bounds for the PGF values on the contour, and establish the relationship between the differentiability of the offspring PGF and the final size PGF as a function of parameters that control the offspring distribution. Finally, we provide demonstrations using simulated and real-world data, and discuss potential identifiability issues regarding statistical inference that is associated with likelihoods that are not conditioned on extinction.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

Coupled gradient descent - where the update of one parameter depends on another - arises naturally in bilevel optimization, two-time-scale stochastic approximation, and generative adversarial networks. When the coupled Jacobian is block-triangular, asymptotic stability is determined by the spectral radii of the diagonal blocks, yet transient amplification before convergence can be arbitrarily large due to non-normality. We develop a sharp pseudospectral theory for block-triangular Jacobians J = [[A, 0], [C, D]], proving Kreiss-constant bounds of the form K(J) <= 2/(1-\gamma) + ||C||/(4(1-\gamma)) when \rho(A), \rho(D) <= \gamma < 1 and A, D are symmetric, and establishing matching minimax lower bounds. We characterize the critical coupling threshold for spectral instability and extend the theory to nearly self-referential systems via a Neumann-series perturbation framework. As a consequence, we obtain a finite-horizon O(K(J)^2 log(1/\delta)) iteration complexity bound. Framed as scaling laws for stochastic two-time-scale optimization, our results expose a non-asymptotic, instance-dependent regime of high-dimensional learning dynamics that is invisible to spectral-radius analysis. Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.

Gradient observations can substantially improve Gaussian process (GP) surrogates, particularly in high-dimensional settings where function evaluations are expensive. However, exact inference with $n$ function values and $n$ full gradients in $d$ dimensions scales cubically in the joint state size, imposing an intractable $\mathcal{O}(n^3 d^3)$ computational bottleneck. We introduce TERA, a highly scalable derivative GP method based on target-specific exact gradient reduction. We prove that for stationary kernels, the gradient components orthogonal to the directions connecting the target and conditioning points are conditionally independent of the target function value; consequently, the exact conditional density is fully characterized by at most $m^2$ directional derivatives once a conditioning set of size $m$ is specified. By using these reduced, dimension-free conditionals as local factors in a Vecchia approximation, TERA effectively decouples $n$ and $d$ from the dense matrix inversion. This reduces the per-target evaluation cost to $\mathcal{O}(dm^2 + m^6)$ time and $\mathcal{O}(dm^2 + m^4)$ memory, leaving the underlying derivative GP model mathematically unchanged. Empirical evaluations demonstrate that TERA achieves state-of-the-art predictive accuracy while operating orders of magnitude faster than standard derivative GPs. Crucially, both computation time and peak GPU memory remain essentially flat with respect to $d$, enabling highly scalable inference in high-dimensional spaces.

Neural posterior estimation (NPE) is a simulation-based approach to Bayesian inference that trains a neural network to approximate the posterior distribution from simulated parameter - data pairs, bypassing likelihood evaluation. We apply NPE – to our knowledge for the first time – to stochastic susceptible-infectious-removed (SIR) epidemic models observed through final outcome data, considering both homogeneously mixing and household-structured populations. Such data arise naturally in retrospective outbreak investigations and household transmission studies, yet inference is computationally challenging: data-augmentation Markov chain Monte Carlo (MCMC) can be slow to mix in large populations and difficult to implement, while Approximate Bayesian Computation (ABC) suffers from low acceptance rates, particularly for large populations or unlikely outcomes. The discrete, low-dimensional nature of such observations makes this setting particularly well suited to NPE. We show that a logNormal posterior approximation, parameterised by a feed-forward neural network, accurately recovers reference posteriors across a range of population sizes and transmission regimes, and extends naturally to joint inference on global and local transmission rates in the household model. Once trained, the network produces approximate posterior distributions in seconds and generalises reliably to population sizes and structures not seen during training. Performance on both synthetic and real outbreak datasets is consistently strong, with results in close agreement with published analyses.