Papers

In deep networks with small initialization, training can exhibit long plateaus separated by sharp feature-acquisition transitions. Whereas shallow nonlinear networks and deep linear networks are well studied, extending these analyses to deep nonlinear networks remains challenging. We derive exact scalar and matrix identities for the imbalance of layer weight norms, holding for any smooth activation and any differentiable loss, and use the resulting approximate balance law to control the full finite-width gradient flow through its first escape. This gives a scaling law in the escape time: with a prefix of $r$ layers initialized at the bottleneck scale $\varepsilon$, the escape time is $\Theta(\Gamma^{-1})$ for $r=1$, $\Theta(\Gamma^{-1}\log(1/\varepsilon))$ for $r=2$, and $\Theta(\Gamma^{-1}\varepsilon^{-(r-2)})$ for $r\geq3$, where $\Gamma=\|\mathbb{E}[yx]\|$. We further show that a multimode teacher leaves the saddle once rather than in stages: with scalar output, all mode overlaps grow through a common amplitude, with relative weights fixed in advance. We find close agreement between our theory and experiment.

Standard Ornstein-Uhlenbeck (OU) models often yield biased parameter estimates when measurement error is ignored. While the Ornstein-Uhlenbeck State Space Model (OUSSM) addresses this in univariate settings, multidimensional extensions remain limited. This paper introduces the factor OUSSM to model multi-dimensional, mean-reverting systems with observational noise. We resolve critical identifiability challenges in parameter estimation by establishing necessary constraints and validating the method through extensive simulations. We demonstrate the model's versatility by analyzing human gut microbiome dynamics and North Atlantic Sea Surface Temperature (SST) data. The results reveal distinct latent temporal structures in both biological and environmental systems, establishing the factor OUSSM as a robust framework for multivariate time series analysis.

Total variation denoising (TVD) is a classical method for denoising and curve fitting, yet an explicit pointwise description of its fitted values has only recently been established in the mean regression setting by arXiv:2410.03041v4. This raises the question of whether a similar representation holds for quantile regression. We answer this question affirmatively by deriving an exact minmax/maxmin representation for the quantile TVD estimator, providing a complete pointwise characterization of its solution set. Given that the quantile TVD estimator is generally non-unique, the existence of such a representation is perhaps surprising. We show that the set of admissible fitted values at any location forms a compact interval, whose endpoints are characterized exactly by minmax/maxmin functionals of local order statistics over nested intervals. We next develop several structural properties of the quantile TVD solution set. First, the solution set is closed under coordinatewise maximum and minimum, guaranteeing the existence of extremal elements – upper and lower envelope solutions. Second, this reveals that quantile TVD is intrinsically non-crossing across quantile levels when a common tuning parameter is used. We prove this is driven by submodularity of the total variation penalty, and show that any penalized quantile regression estimator with a submodular penalty enjoys this property. From an estimation error perspective, our representation enables a refined pointwise analysis via a transparent local bias-variance decomposition, facilitating new pointwise risk bounds and near-optimal rates for locally Holder smooth functions. Our results hold under heavy-tailed noise (e.g., Cauchy) and substantially extend existing guarantees beyond locally constant signals. Altogether, these results advance the theory of quantile TV regression via exact pointwise min-max representations.

We develop a modular approach to Markov chain Monte Carlo (MCMC) sampling for unnormalized target densities. In this approach, Markov chains are constructed in parallel, each constrained to a subset of the target space. The Monte Carlo estimates from the constrained chains are then combined with appropriate weights, calculated from the transition probabilities between subsets. In addition to the computational advantages arising from its parallelized structure, this modular MCMC approach enables variance reduction for Monte Carlo estimation in settings where sampling from low-density regions is required. We develop a central limit theorem-type result for the resulting Monte Carlo estimates and propose a method for estimating their standard errors. Furthermore, by applying this modular sampling technique to simulated tempering, we propose a method for Monte Carlo estimation of expectations with respect to multimodal target distributions. This approach effectively addresses a well-known challenge of tempering-based methods: sampling efficiency can be greatly reduced when separated modes of the target distribution have different scales. We demonstrate the efficiency of the proposed methods through numerical examples, including one arising from Bayesian sparse regression with a spike-and-slab prior.

Log-logistic distribution is a flexible distribution that can model a wide range of failure patterns in the field of electrical, electronic and mechanical engineering and is often used in reliability inference. However, the inference of the parameters and reliability function of the log-logistic distribution can be challenging, especially in small sample scenarios. In this paper, we propose a new inference framework based on the least squares estimator-based generalized pivotal quantities (LSE-GPQ) for the parameters and reliability functions of the log-logistic distribution, which can provide better coverage in small sample scenarios. We will compare the performance of our proposed method with traditional methods such as the MLE and parametric bootstrapping through simulation studies and real data applications.

We present a non-asymptotic theory of generalization in deep learning where the empirical neural tangent kernel partitions the output space. In directions corresponding to signal, error dissipates rapidly; in the vast orthogonal dimensions corresponding to noise, the kernel's near-zero eigenvalues trap residual error in a test-invisible reservoir. Within the signal channel, minibatch SGD ensures that coherent population signal accumulates via fast linear drift, while idiosyncratic memorization is suppressed into a slow, diffusive random walk. We prove generalization survives even when the kernel evolves $\mathcal{O}(1)$ in operator norm, the full feature-learning regime. This theory naturally explains disparate phenomena in deep learning theory, such as benign overfitting, double descent, implicit bias, and grokking. Lastly, we derive an exact population-risk objective from a single training run with no validation data, for any architecture, loss, or optimizer, and prove that it measures precisely the noise in the signal channel. This objective reduces in practice to an SNR preconditioner on top of Adam, adding one state vector at no extra cost; it accelerates grokking by $5 \times$, suppresses memorization in PINNs and implicit neural representations, and improves DPO fine-tuning under noisy preferences while staying $3 \times$ closer to the reference policy.

Conventional algorithmic trading systems are grounded in deterministic heuristics or offline-trained statistical models that cannot adapt to the semantic complexity of rapidly shifting market regimes. This paper introduces AGENTICAITA, an agentic AI framework that replaces the traditional signal then execute paradigm with a fully autonomous deliberative loop in which multiple specialized Large Language Model agents reason, negotiate, and act in concert - without any offline training or human intervention. The framework proposes four architectural contributions: (i) an Adaptive Z-Score Trigger Engine that acts as a cognitive resource allocator, gating LLM inference exclusively on statistically anomalous market conditions; (ii) a Sequential Deliberative Pipeline - the core agentic contribution - in which an Analyst agent, a Risk Manager agent, and an Executor agent form a structured reasoning chain governed by typed JSON contracts and a deterministic hard-gate safety layer; (iii) an Inference Gating Protocol, a mutex-based cognitive resource scheduler that serializes concurrent agent activations and ensures fully reproducible audit trails; and (iv) a Correlation-Break Diversification composite score that operationalizes portfolio-level idiosyncratic signal prioritization within individual agent reasoning. Validated over a five-day autonomous dry-run session under live market conditions, the framework demonstrates operational correctness of the deliberative pipeline, achieving 157 zero-intervention invocations across 76 assets with an 11.5% agentic friction rate that confirms non-trivial inter-agent negotiation. This preliminary proof-of-concept establishes the feasibility of training-free, deterministic safety-constrained multi-agent orchestration in financial decision loops, with statistically robust performance evaluation and execution cost modeling deferred to extended live deployment.

In American options, the early exercise feature allows the option to be exercised at any time prior to expiration. However, this flexibility introduces a challenge: the pricing model must value the option while simultaneously determining an unknown, time-varying exercise boundary. The Heston model is one of the most popular ways to model real market behavior because it allows volatility to change over time. However, unlike European options, there is no closed-form solution for American options under the Heston model, so we have to use numerical methods. In this paper, we propose a novel approach to solving the stochastic Heston partial differential equation for American options, using coupled physics-informed neural networks (PINNs) to predict both the option price and the free boundary, while employing curriculum learning and adaptive resampling to stabilize model training. Our work builds on recent deep learning methods but introduces a more effective training strategy to address the limitations of these approaches. The numerical results demonstrate the effectiveness of the proposed learning framework, providing a robust and efficient alternative to pricing American options, enabling rapid inference and accurate estimation under stochastic volatility.

We propose a novel method of understanding disease transformation from a healthy baseline with biomarker-level explainability. By modeling the biomarker covariance matrices of healthy controls and disease states, the perturbation can be individually characterized to accomplish mechanistic explanations of disease trajectories, both at a molecular level and for individual patients. Given a cohort of n patients each measured on p biomarkers, we define the biomarker "Hamiltonian" H = X^T X / n \in R^{p \times p}, where X \in R^{n \times p} is the covariant biomarker matrix. The eigenvectors of H define a set of normal modes of biomarker coordination, and the eigenvalues quantify the energy carried by each mode. In the healthy state, the reference Hamiltonian H_0 governs this structure where disease perturbs H_0 by an additive operator \Delta H, thus shifting eigenvalues and rotating eigenvectors in proportion to the severity of pathological disruption. We formalize this framework, derive the spectral change given a disease perturbation, and demonstrate that the projection of a newly diagnosed patient's cumulative biomarker covariance structure onto disease-discriminant eigenmodes constitutes an optimal prognostic statistic for greater precision in disease prognosis. This work serves as a veritable white paper with application across a panoply of disease frameworks from cancer to neurodegenerative disorders.

We develop a coarse-to-fine generalized linear mixed model (CF-GLMM) for scalable spatial prediction of exponential-family responses. CF-GLMM extends coarse-to-fine spatial modeling (CFSM) beyond Gaussian data by sequentially increasing spatial resolution while validation deviance improves, so the terminal resolution need not be specified in advance. The latent spatial process is constructed by aggregating local models, avoiding costly operations on large spatial covariance matrices. An idealized convolution representation links the covariance of our assumed spatial process to a sum of Matern covariances with different ranges. Monte Carlo experiments show that CF-GLMM achieves predictive accuracy comparable to widely used scalable spatial GLMMs while requiring less computation and memory. Its scale-specific components also support exploratory multiscale analysis. An application to COVID-19 infection data in Tokyo illustrates fine-resolution spatial prediction and multiscale characterization of geographical variation. The proposed method is implemented in an R package spCF (https://cran.r-project.org/web/packages/spCF/).

Spectral graph sparsification is a classical tool for reducing graph complexity while preserving Laplacian quadratic forms. In graph neural networks (GNNs), sparsification is often used to accelerate computation while maintaining predictive performance. In this work, we study a complementary representation-level question: does sparsification preserve the geometry of learned embeddings? For polynomial-filter GNNs, we prove that any $\epsilon$-spectral sparsifier induces $O(\epsilon)$ perturbations in polynomial graph filters, multilayer hidden representations, and their Gram matrices. These guarantees imply stability of squared pairwise distances, class means, and covariance structure in embedding space. We further establish finite-time training stability: under smoothness and boundedness assumptions, gradient descent on dense and sparsified graphs produces weight trajectories whose separation grows at most proportionally to the sparsification distortion. Empirically, effective-resistance sparsification validates the predicted perturbation chain on synthetic graphs and preserves hidden representation geometry on real datasets. In our experiments, the gram matrix and training dynamics show low divergence even under substantial sparsification, consistent with the predicted stability under spectral sparsification. Hidden Gram preservation strongly predicts neighborhood preservation and class-centroid stability across FashionMNIST, Cora, and Paul15. Together, these results show that spectral sparsification preserves not only graph operators, but also the representation geometry that supports downstream use of GNN embeddings for interpretability.

The traditional kernel density estimator of an unknown density is by construction completely nonparametric, in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.

Difference-in-differences (DiD) identification relies mainly on a parallel trends assumption about untreated potential outcomes. Researchers often relax this assumption by assuming conditional parallel trends within units with the same covariate values. However, the process of selecting which covariates to include in this assumption is often ad hoc. We propose a formal approach to select the variables that support conditional parallel trends based on graphical criteria. We show that the parallel trends assumption is rarely justified without conditioning on covariates, and that unconditional and conditional parallel trends can conflict with one another. We also demonstrate that a time-invariant covariate with a time-invariant effect on the outcome, which might not ordinarily be considered a confounder in DiD, may be a useful conditioning variable. We clarify that adjustment for a post-treatment covariate depends on what causes that covariate to change. Extending our framework to multiple time periods, we distinguish between treatment type and rollout strategy and examine the problem of treatment-confounder feedback. On the estimation side, we argue that the difficulty of incorporating covariates in DiD, often framed as an estimator problem, is more accurately understood as a misalignment between the adjustment set used by the estimator and the adjustment set required for identification. This misalignment affects several popular estimation procedures, and resolving it requires not a change of estimator, but a change in how covariates enter the estimation procedure. We show how to achieve this alignment for all estimators we evaluate.

Graph neural tangent kernels give a principled infinite-width theory for graph neural networks, but inherit a basic limitation of graph models: they see only pairwise structure. Many relational systems contain higher-order interactions that are more naturally represented by simplicial complexes. We introduce the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing on edge features. TopoNTK combines lower Hodge interactions, capturing graph-like coupling through shared vertices, with upper Hodge interactions, capturing coupling through filled simplices. This makes the kernel sensitive to topology invisible to graph kernels, allowing complexes with the same graph but different filled simplices to induce different kernels. Beyond expressivity, the Hodge structure gives the kernel an interpretable learning geometry. Edge signals decompose into gradient-like, harmonic, and local circulation components, and the spectrum of the TopoNTK determines how quickly each component is learned. This yields a topological form of spectral bias: components aligned with large-eigenvalue modes are learned quickly, while global harmonic modes, retained through the residual channel, often lie at smaller eigenvalues and are learned more slowly. We prove expressivity, Hodge-alignment, spectral learning, and stability properties, and validate them on synthetic simplicial tasks and DBLP higher-order link prediction. The results show that topology is not merely extra structure; it can provide coordinates that make relational learning more faithful, interpretable, and effective.

Feedforward neural networks transform data through learned representations whose geometry shapes how classes separate and relate across successive layers. We study that geometry through diffusion operators. Each feature-cloud snapshot is assigned a Gaussian-kernel Markov operator, giving a smooth description of one-step transport between classes from which spectral, boundary, and local geometric information can be read. We define the empirical class chain, state the condition under which it is an exact Markov quotient, and derive both the corresponding population transition and a simpler overlap chain based on expected class affinities. For balanced shared-covariance Gaussian class-conditional snapshots these affinities have closed forms controlled by a regularized Mahalanobis separation, which yields explicit expressions for leakage and coarse spectral behaviour. We further show that operator observables vary smoothly under feature perturbations, whereas hard neighborhood graphs are controlled by neighbor-order margins. Experiments on CIFAR-10 and CIFAR-100 ResNet-18 representations find that class transport becomes increasingly persistent with depth while retaining structured relations between classes, and that the diffusion class chain is more stable than its $k$-nearest-neighbor counterpart under matched perturbations.

The ensemble Gaussian mixture filter (EnGMF) is a powerful, convergent particle filter capable of medium-to-high dimensional non-linear filtering. The EnGMF relies on a resampling step that can generate physically unrealistic posterior samples, that would subsequently produce physically meaningless forecasts. This work introduces the discriminator-informed resampling procedure, that augments the posterior resampling step with a discriminator that accepts or rejects candidate particles based on their physical plausibility. In this work these discriminators are learned through a normalizing flow approach. Numerical experiments on both the Ikeda map and the Lorenz '63 system show that discriminator informed resampling procedure consistently reduces error relative to the standard EnGMF in low-ensemble regimes.

In this paper, we develop and analyze a nonparametric procedure for detecting a single change point in sequences of independent observations using energy distance. The asymptotic properties of the test statistic are derived under both null and alternative hypotheses. Under the null hypothesis, for any fixed candidate split point, the standardized statistic $\mathcal{Z}_{n,k}$ converges to a standard normal limit. For global detection, we use the scan statistic $T_n=\max_{k\in K_\eta}|\mathcal{Z}_{n,k}|$ and calibrate critical values using a permutation test, which yields valid type I error control under exchangeability. The simulation study shows that the proposed method demonstrates much better robustness across various error distributions. To handle multiple change points in practical applications, the method is combined with a binary segmentation approach. The breast cancer cell line (MDA157) from cDNA microarray CGH data is used to illustrate the detection and estimation capabilities of the proposed method for genomic sequences.

We show that time-reversed Young interferometry reorganizes, rather than reverses, optical entropy. A fixed detector conditions the reciprocal source–detector Green function and produces a source-label probability distribution. Marginal entropies in the standard and time-reversed geometries are generally unequal; the reciprocal invariant is instead the mutual information between source and detector coordinates. Near a destructive response, the conditioned source-label entropy can decrease while Fisher information for small phase, tilt, or defocus perturbations increases. The result identifies time-reversed Young interferometry as a source-space information processor with no analogue in ordinary detector-plane fringe readout.

Clinical trials usually target average treatment effects, but treatment decisions are made for individuals. This tension motivates a common criticism of evidence-based medicine: a treatment that is beneficial on average may be inappropriate for a particular patient, and skilled physicians may outperform rigid adherence to the strategy that performed best in a randomized trial. We consider how randomized and observational data from the same target population can be used to assess that possibility. Specifically, we study settings in which trial and/or observational data yield estimates of average outcomes under `treat all', `treat none', and usual care strategies in the population of interest. We derive sharp bounds on the proportion of encounters with physicians whose personal strategies outperform the better of `treat all' or `treat none' under the assumption that no physician's strategy is worse than always choosing the worse performing of `treat all' and `treat none'. These results clarify when clinical data support relying on physician discretion over the trial-average recommendation and when stronger justification is required.

Statistical change point (CP) detection methods typically rely on likelihood-based inference and ignore contextual information about plausible CP locations beyond the observed sequence. Although informative priors provide a natural way to incorporate such information, general and computationally efficient methods for doing so are lacking, especially for multiple CP detection. To address this gap, we propose a prior-informed CP detection algorithm (Pi-Change) that incorporates prior information on CP locations through a time-varying penalty term. We prove that the proposed penalty can be embedded in the Pruned Exact Linear Time framework while preserving the dynamic programming recursion and pruning rule required for efficient multiple CP detection. Across simulation studies and three time-series applications, Pi-Change discourages spurious CPs unsupported by prior information, remains robust to prior misspecification, and improves detection accuracy. More broadly, Pi-Change extends multiple CP detection beyond purely data-driven fitting by incorporating partial prior knowledge in a computationally efficient and interpretable way. It is particularly useful when CPs arise from heterogeneous mechanisms or are associated with known external events, helping quantify the delay between an event and the resulting structural change.

(Abridged) Simulation-based inference (SBI) has emerged as a powerful framework for extracting cosmological information from complex, non-linear data where analytical likelihoods are unavailable. Its reliability is commonly assessed using coverage-based diagnostics under the prior predictive distribution, which probe calibration only in an averaged sense and do not constrain posterior behavior at fixed parameter value, the regime relevant for practical inference. We investigate these limitations in the context of primordial non-Gaussianity, parameterized by $f_\mathrm{NL}$, using simulations of the dark matter halo field. We compare SBI based on contrastive neural ratio estimation (CNRE) with likelihood-based inference (LBI) using the power spectrum, bispectrum, and wavelet scattering transform (WST) coefficients across 1000 realizations. SBI and LBI agree well on posterior means and skewness, while the variance agrees on average but shows weaker realization-by-realization consistency. Larger differences arise in the kurtosis, indicating discrepancies in the posterior tails. These effects are already present for the power spectrum - where the Gaussian likelihood assumed in LBI is best justified - and are most pronounced for the combined power spectrum and bispectrum, where SBI posteriors are often underconfident and can yield weaker constraints than either statistic individually, despite passing coverage tests. WST coefficients further tighten constraints on $f_\mathrm{NL}$, even when restricted to large scales. Our results highlight both the potential of higher-order statistics and the need for validation strategies that probe the posterior shape beyond standard coverage diagnostics.

Hierarchical Gaussian Filtering (HGF) networks allow for efficient updating of posterior distributions (beliefs) about hidden states of an agent's environment. HGF parent nodes can target the mean or variance of their children. New information entering at input nodes leads to a cascade of belief updates across the network according to one-step update equations for each node's mean and precision (inverse variance). However, the original form of the update equations for variance-targeting parents(volatility coupling) can in some regions of parameter space lead to negative posterior precision, a logical impossibility which causes the updating algorithm to terminate with an error. In this report, we introduce a modified quadratic approximation to the variational energy of volatility-coupled nodes that avoids negative posterior precision. The key idea is to interpolate between two quadratic expansions of the variational energy: one at the prior prediction and one at a second mode whose location is obtained in closed form via the Lambert W function. The resulting update equations are robust across the entire parameter space and faithfully track the variational posterior even for large prediction errors.

Coherent Point Drift (CPD) is a representative probabilistic framework for unsupervised non-rigid point set registration. Its standard non-rigid M-step, however, relies on a point-indexed Gaussian-kernel system whose size grows with the number of moving points, making deformation estimation computationally heavy for large point sets and difficult to control in complexity during registration. To address these limitations, we propose Analytic-CPD, a new unsupervised non-rigid registration framework that gives CPD a structured analytic reformulation. Analytic-CPD preserves the CPD posterior correspondence layer, but lifts the M-step from point-indexed kernel displacement estimation to structured analytic mapping estimation. By coupling the Gaussian-mixture posterior mechanism of CPD with Structured Analytic Mappings (SAM), the method obtains a deformation model whose coefficient dimension is governed by the ambient dimension and analytic order rather than by the number of moving points. More importantly, deformation estimation is organized over an interpretable hierarchy of analytic function spaces, so the analytic order can be increased progressively as posterior correspondences become more reliable. We implement this idea through an increasing-degree continuation strategy with decreasing stage lengths: low-order analytic maps first stabilize the posterior correspondence structure, while higher-order modes later refine nonlinear residual deformation. Experiments on controlled model-matched, smooth model-mismatch, and registered human-shape data demonstrate the effectiveness and favorable accuracy–efficiency performance of Analytic-CPD.

High-dimensional vector autoregressive (VAR) models provide a flexible framework for characterizing dynamic dependence in multivariate spatio-temporal systems, but their unrestricted estimation becomes infeasible when multiple variables are observed over many spatial locations. This paper develops a structured estimation procedure for high-dimensional multivariate VAR processes that explicitly incorporates spatial information. We decompose each block transition matrix into a cross-variable dependence coefficient and a spatial transition matrix, and constrain the spatial transition matrices through a pre-specified spatial graph. The resulting estimator is formulated as a weighted $\ell_1$-regularized least-squares problem, where the weights encode spatial proximity or topological similarity and induce stronger shrinkage on spatially implausible interactions. Since the objective function is bi-convex, we estimate the cross-variable dependence matrix and the spatial transition matrices through an alternating convex-search algorithm implemented with ADMM. Under stability and restricted-eigenvalue-type conditions for high-dimensional VAR processes, we establish convergence to a blockwise stationary point in the subgradient sense and derive high-probability estimation error bounds for both components of the model. Simulation studies demonstrate that the proposed estimator accurately recovers sparse transition structures and improves over existing two-step $\ell_1$-regularized methods in support recovery and estimation accuracy. An application to North American climate data illustrates that the method recovers interpretable variable-dependence networks and spatial interaction patterns across different climate regions.

We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh–Nagumo neurons, and a stochastic Kuramoto model.

This study evaluates the performance of 36 historical CMIP6 GCM trajectories (1979-2005) in reproducing atmospheric circulation over the Iberian Peninsula in the summer months (June-September) using the Lamb Weather Type (WT) classification scheme. Using ERA5 reanalysis as the observational reference, we introduce a methodological framework-applicable to any region worldwide-to evaluate GCM performance. This approach extends traditional daily frequency analysis by evaluating both the daily frequency distribution of WTs and their 24-hour dynamic evolution (i.e., transition probabilities and persistence). Model performance is quantified using the Overlap coefficient. A filtering process is applied where only trajectories that successfully reproduce both daily and conditional distributions with a minimum Overlap threshold $t_{sim}$ across a set number of grid points are retained. The findings show that while several models can adequately reproduce daily WT frequencies (16 out of 36), some struggle to capture day-to-day atmospheric transitions. This leads to a final selection of 12 trajectories over the Iberian Peninsula. Model performance across the region is then evaluated using integrated metrics assessing daily reproduction, conditional reproduction, and transition dynamics. Overall, models from the ec earth3 family-specifically the ec earth3 aerchem trajectory-exhibit the best and most consistent performance across the region. Additionally, the results highlight a geographical performance gap: while models generally represent circulation well in the northwest, they face significant challenges in the central and southern Mediterranean regions of the Peninsula. Ultimately, this study establishes that assessing WT persistence and transitions provides a far more discriminative, objective tool for GCM selection than evaluating daily distributions alone.

Estimating network formation models with degree heterogeneity raises two problems in empirical networks. First, agents that send no links, receive no links, or link to all remaining agents can make the fixed-effects MLE fail to exist. Trimming these agents changes the estimation sample and induces selection bias. Second, the incidental-parameter problem biases common parameters and average partial effects. We resolve both issues through a penalized likelihood approach. Our leading specification is a directed network model with reciprocity, nesting the standard undirected and non-reciprocal directed models. The penalty guarantees finite-sample existence and yields bias corrections for coefficients and partial effects. We establish asymptotic results without imposing compactness on the fixed-effects. Allowing the fixed effects to diverge at a logarithmic rate, our asymptotic framework accommodates the degree sparsity ubiquitous in large empirical networks. A global trade application demonstrates that our estimator avoids selection bias and recovers robust parameters where conventional methods fail.

We propose a computationally efficient inferential procedure for longitudinal function-on-function regression. The method follows a marginal three-step approach: (1) fit massive pointwise longitudinal scalar-on-function regression models, (2) smooth the resulting estimates along the bivariate functional domain, and (3) compute confidence bands using either an analytic approach for Gaussian data or a cluster bootstrap for Gaussian or non-Gaussian data. Simulation studies demonstrate that the proposed method achieves accurate estimation and valid inference, while substantially reducing computational burden compared to existing approaches. Methods are motivated by a physical activity intervention trial in older adults where high-dimensional wearable data were collected longitudinally across multiple visits. Our applications reveal significant increases in physical activity in the morning using interpersonal intervention strategies, but not intrapersonal strategies. The proposed methods are implemented in an R package.

Networked systems operating under intermittent adverse conditions and long memory can remain stable on average while exhibiting rare but extreme trajectory-level excursions. We study linear regime-switching network dynamics with Volterra-type memory, formulated through a finite-dimensional lifted ordinary differential equation embedding. Despite finite-horizon annealed boundedness, we show that quenched amplification emerges generically from the interaction of regime persistence, memory accumulation, and non-normal lifted operator geometry. A lower bound on burst-size distributions reveals power-law tails whose exponent is determined by the ratio between unfavorable dwell-time rates and an operator-defined instantaneous growth parameter. This parameter is computable online via the Euclidean logarithmic norm of the lifted operator, yielding a practical early-warning indicator. Building on this structure, we introduce a dynamic data-driven intervention strategy that enforces contraction on demand along rare amplification channels, thereby shaping or truncating tail risk without altering exogenous regime statistics or typical system behavior. The results provide a geometrically grounded and operationally actionable framework for understanding and mitigating extreme events in memory-driven regime-switching systems.

LLMs excel at predictive tasks and complex reasoning tasks, but many high-value deployments rely on decisions under uncertainty, for example, which tool to call, which expert to consult, or how many resources to invest. While the usefulness and feasibility of Bayesian approaches remain unclear for LLM inference, this position paper argues that the control layer of an agentic AI system (that orchestrates LLMs and tools) is a clear case where Bayesian principles should shine. Bayesian decision theory provides a framework for agentic systems that can help to maintain beliefs over task-relevant latent quantities, to update these beliefs from observed agentic and human-AI interactions, and to choose actions. Making LLMs themselves explicitly Bayesian belief-updating engines remains computationally intensive and conceptually nontrivial as a general modeling target. In contrast, this paper argues that coherent decision-making requires Bayesian principles at the orchestration level of the agentic system, not necessarily the LLM agent parameters. This paper articulates practical properties for Bayesian control that fit modern agentic AI systems and human-AI collaboration, and provides concrete examples and design patterns to illustrate how calibrated beliefs and utility-aware policies can improve agentic AI orchestration.