Papers

While algorithmic stability is a central tool for understanding generalization of learning algorithms, existing high-probability guarantees typically rely on uniform boundedness or sub-Gaussian/sub-Weibull tail assumptions, which can be overly restrictive for modern settings with heavy-tailed or unbounded losses. We develop a stability-based framework that requires only a finite $L_p$ moment condition. Our first contribution is sharp concentration inequalities for functions of independent random variables under $L_p$ constraints, extending McDiarmid's bounded-differences techniques beyond the classical regime. Leveraging these results, we derive sharp high-probability generalization bounds across a range of learning paradigms, including empirical risk minimization, transductive regression, and meta-learning. These guarantees show that $L_p$ stability suffices for robust generalization even when boundedness fails, substantially weakening the standard assumptions in the stability literature.

The transformer's emergent ability to perform in-context learning (ICL) has sparked a wide range of studies designed to understand its underlying mechanisms. Existing works often study how training task diversity, defined either as the number of ICL training task vectors or as the number of function classes from which the task vectors are drawn, shapes both the learning dynamics and generalization capabilities of ICL. While both definitions have uncovered many interesting phenomena, many observations under the latter definition remain theoretically unexplained. This paper presents a minimal analytical model under which these phenomena provably emerge from the properties of the training data. By modeling the training task vectors as a mixture of low-rank Gaussians, we show how training task diversity, defined by the number of non-overlapping columns between subspaces that parameterize the covariance matrices, improves both the generalization and optimization trajectory of ICL with linear attention. In particular, we show that our model can explain (i) why training with task diversity shortens the ICL plateau and (ii) why ICL appears to achieve out-of-distribution generalization. We conclude by empirically demonstrating how our results extend to nonlinear transformers and nonlinear function classes. Overall, our work presents a tractable framework to unify existing observations.

Digital learning environments record learners' responses to individual items, making it possible to study the development of specific skills rather than overall scores. Drawing conclusions about learning from these data requires a model that links responses to latent skills and tracks how mastery changes over time. When the skills measured by each item are unknown, the analyst must decide whether to estimate this structure, the Q-matrix, jointly with the learning process, or to establish it first and study learning afterwards. We show that this decision can change substantive conclusions about how learners develop. Using dynamic cognitive diagnostic models, we analyse data from two reading games measuring vocabulary and comprehension from Grade 2 to Grade 3, with item-text embeddings providing prior information for the unknown Q-matrix. A joint analysis and a bias-corrected stepwise analysis agree that most learners move toward mastering both skills, but disagree about how many remain only partially proficient at Grade 3, changing how reading progress would be reported. A simulation study identifies when the two analyses diverge and shows that joint analysis is more reliable when the item-skill structure is uncertain and the item pool changes between grades. We provide R code for both analyses.

We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution \(\widehat{\pi}_{0,\rho}\). For each later sliding window, we compute \(\widehat{\pi}_{t,\rho}\) and define the score \[ S_t=\|\widehat{\pi}_{t,\rho}-\widehat{\pi}_{0,\rho}\|_1. \] Large values of \(S_t\) indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.

Relational Event Models (REMs) provide a rigorous framework for analyzing dyadic interactions observed in continuous time, capturing history-dependent dynamics such as triadic closure and reciprocity. Framing REMs through the lens of counting processes embeds the model in a rich theoretical foundation, facilitating its mathematical analysis. While Maximum Likelihood Estimation (MLE) is standard practice for estimating these models, the underlying statistical guarantees rely on specific asymptotic regimes, namely, whether the network size (n), the observational period (T), or both approach infinity. We review the theoretical foundations of such counting-process-based models, formalizing the core assumptions required to achieve asymptotic normality across these different limits. With a specific focus on Cox-type multiplicative models, we detail the circumstances under which these assumptions hold. Supported by simulation studies, we illustrate how structural modeling choices, including temporal windowing and logarithmic transformations, affect empirical coverage and estimator convergence. We thereby derive several guiding principles for specifying such models in realistic contexts, bridging theory and practice.

Electronic health records (EHR) pose large-scale multi-disease modeling problems in which many outcomes are rare and strongly influenced by shared risk factors. While modern approaches achieve strong predictive performance, they often treat diseases independently or rely on black-box architectures, offering limited insight into how risk factors organize disease risk and little principled uncertainty quantification. We introduce a Bayesian hypergraph inference framework that reframes multi-disease modeling around latent, risk-factor-modulated disease pathways. Risk factors act on hyperedges, latent disease subsets with shared risk patterns, allowing diseases to participate in multiple distinct pathways and enabling interpretable, higher-order structure beyond pairwise associations. A repulsion prior encourages parsimonious and identifiable structure, while posterior inference provides calibrated uncertainty over both disease groupings and risk-factor influence. To enable scalable inference on large EHR datasets, we develop a structured variational inference algorithm that preserves logical dependencies among hyperedge existence, disease membership, and pathway-level effects. Experiments on simulated data and UK Biobank demonstrate stable and interpretable disease pathway structure, well-calibrated uncertainty, improved estimation for rare diseases, and competitive predictive performance.

Robust Subspace Recovery (RSR) aims to identify an underlying d-dimensional subspace from a dataset heavily corrupted by outliers. Complexity-theoretic results establish a threshold for the problem's computational hardness based on the dimension-scaled signal-to-noise ratio (DS-SNR): the problem is SSE-hard when the DS-SNR is strictly less than 1, and solvable via practical algorithms when it is greater than 1 under general position assumptions. However, the exact behavior of practical algorithms at the critical boundary DS-SNR = 1 has remained unknown. This work resolves the behavior of Tyler's M-estimator (TME) at this critical boundary, consequently establishing a sharp phase transition. Specifically, we prove that TME converges exactly to the true subspace for DS-SNR \geq 1 under a new stability condition, which is less restrictive than the general position assumptions used in prior literature. Our analysis utilizes a decomposition of the TME iterates within a majorization-minimization framework.

Characterizing the optimization dynamics and statistical performance of over-parameterized deep neural networks (DNNs) remains a central challenge in understanding the remarkable success of deep learning. We establish quantitative bounds showing that kernel gradient descent in the reproducing kernel Hilbert space induced by the deterministic infinite-width neural tangent kernel approximates finite-width deep regression with smooth activations under gradient descent (GD) and stochastic gradient descent (SGD) training. The approximation gap is governed by the network width and training horizon, with an additional stochastic gradient error in the SGD case. This connection provides a general mechanism for transferring learning-theoretic guarantees from kernel methods to deep regression. As an application, under general source and effective dimension conditions, we show that both GD- and SGD-trained DNNs attain the minimax-optimal excess population risk rate, up to logarithmic factors, provided that the network width grows polynomially in the sample size. To the best of our knowledge, these are the first such guarantees for standard fully connected deep neural networks with smooth activations trained by GD and SGD.

We explicitly characterise the full class of e-processes for testing conditional non-parametric hypotheses, defined by finitely many conditional constraints. Our main result is a complete-class theorem: every e-process for such a hypothesis is point-wise dominated by a predictable product of affine one-step e-variables. Therefore, for a broad class of conditional testing problems, arbitrary e-processes can be replaced without loss by test supermartingales. This extends previous complete-class results from single-step constrained testing and bounded one-dimensional conditional mean testing to a broader conditional sequential setting.

Recent progress has been made in understanding the statistical generalization performance of gradient descent methods for overparameterized neural networks within the neural tangent kernel (NTK) regime. However, most of the existing work on regression problems is limited to shallow network architectures, leaving a notable gap in the theory of deep neural networks. This paper addresses this gap by presenting a comprehensive generalization analysis for deep ReLU networks trained using gradient descent (GD) and stochastic gradient descent (SGD). Specifically, we establish the first known minimax-optimal rates of excess population risk for both GD and SGD with deep ReLU networks, under the assumption that the network width scales polynomially with respect to the network depth and training sample size. Our results demonstrate that with sufficient width, gradient descent methods for deep ReLU networks can achieve optimal generalization rates on par with kernel methods.

DeepKriging-style models, such as Spatio-Temporal DeepKriging, improve scalability through basis-function embeddings and stochastic gradient learning; however, fixed regular-grid spatial bases remain inefficient under highly non-uniform sampling patterns, often over-allocating capacity to sparse regions while under-resolving dense clusters. To address this limitation, we propose a practical extension of DeepKriging for reliable spatio-temporal distributional forecasting, incorporating cluster-adaptive spatial bases - whose centers and scales are initialized from {the spatial sampling density} - to better capture heterogeneous spatial sampling, together with cluster-aware conformal calibration that determines prediction-interval widths within spatial clusters (with a global fallback when calibration samples are insufficient). The resulting calibration pipeline explicitly targets spatial heterogeneity and local miscalibration, and experiments, including simulation studies and PM$_{2.5}$ data analysis, demonstrate substantially improved coverage accuracy and tail reliability under clustered observation patterns compared with a global conformal baseline.

TorchKM is an open-source library for kernel machines, including support vector machines, kernel logistic regression, and kernel quantile regression, with GPU acceleration. The library features a scikit-learn-style API and is designed to exploit GPU-friendly linear algebra, accelerating the full training and model-selection pipeline through intelligent reuse of matrix operations. Benchmarks show competitive predictive performance with substantial speedups over standard baselines. The efficiency and programmable design also make TorchKM a kernel-learning component for AI-driven workflows. Code and documentation are available at https://github.com/YikaiZhang95/torchkm, and the package can be easily installed via PyPI.

With the increasing availability of ranking data, there has been a growing demand for appropriate unsupervised rank-based inferential frameworks capable of handling high-dimensional datasets and providing uncertainty quantification for all estimates. Rank-based methods have also seen a growing popularity in -omics pipelines, as ranking continuous measurements provides a robust means of handling non-normally distributed data. The Bayesian Mallows model (BMM) has emerged as a promising choice because of its adaptability to various types of ranking data and its flexible framework, integrating cluster-wise rank aggregation with inference at the individual level. However, the scalability of BMM to ultra-high-dimensional settings, such as -omics analyses, has remained limited. The present paper addresses this issue by introducing the first rank-based model generalizing BMM to jointly handle clustering and variable selection, namely the lower-dimensional Bayesian Mallows Model Mixture (lowBM3). The proposed method provides a novel Bayesian framework that simultaneously handles heterogeneity in the sample, unsupervised parameter estimation, and model selection in a scalable manner for ultra-high-dimensional data. Additionally, a companion postprocessing framework is introduced to provide posterior summaries of the discrete posterior distributions of both the consensus ranking and the variable selector. Simulation studies are performed to assess the performance of the method. The usefulness of the method is also shown in an application to signature discovery for cancer genomics, where RNA-seq bulk gene expression data obtained from breast cancer patients are clustered genome-wide.

This paper develops the Dynamic Gaussian Process (DGP), a framework for estimating functions governed by integro-difference equations (IDEs). IDEs model continuous functions that evolve with discrete-time dynamics and arise naturally from time-discretization of linear partial differential equations (PDEs). The DGP extends Gaussian process regression to time-varying functions and extends Kalman filtering to infinite-dimensional states. The DGP posterior remains a Gaussian process with closed-form mean and covariance updates, and separable kernel structure reduces the problem to a finite-dimensional Kalman filter on basis function coefficients. This paper extends the DGP to vector-valued states, enabling the treatment of higher-order PDEs, and provides a stability and approximation error analysis for the basis function approximation. The functional L2 estimation error decomposes exactly into in-subspace and out-of-subspace contributions, and all approximation errors vanish as the number of basis functions grows. The framework is demonstrated on the heat equation and on the wave equation, the latter with a vector-valued state. Code is available at https://github.com/JvHulst/Dynamic_Gaussian_Processes.

Highly reliable products are often tested under accelerated conditions to provoke failures within a feasible timeframe. For products whose service life involves repeated alternation between two stress levels, such as automotive air-conditioners, batteries, and aerospace components, cyclic-stress accelerated life testing (CyALT) provides a more realistic loading profile than conventional accelerated tests. In practice, failures are often recorded only at scheduled inspection times, leading to interval-censored counts rather than exact lifetimes. Moreover, traditional maximum likelihood estimation is sensitive to data contamination, which is a genuine concern in small-sample industrial experiments. This paper develops robust inferential procedures for CyALT models with lognormal lifetimes under interval monitoring. Robust estimators are obtained by minimizing a weighted density power divergence (WDPD), leading to the weighted minimum density power divergence estimator (WMDPDE). We establish the asymptotic distribution of the WMDPDE, derive influence function expressions to characterize the robustness, and present asymptotic and bootstrap confidence intervals for important lifetime characteristics. A simulation study confirms that the WMDPDE provides substantial protection against outliers while retaining high efficiency under clean data. The methodology is illustrated through the analysis of an air-conditioner reliability dataset, demonstrating the practical advantages of robust inference in the CyALT framework.

Routine refresh bundles two operations that need not travel together: estimating parameters and selecting the model form. The second is often unnecessary. Under a reduced-update policy the maintenance question is a stopping problem: when has enough evidence accumulated against the deployed form to justify re-specifying it? We define specification debt as that evidence, derive a cost-sensitive rule that re-specifies when expected avoidable loss exceeds re-specification cost, and recover fixed update frequencies as the constant-accumulation case. The rule then raises its own question: how much machinery the monitor needs. We take this to all 47,982 monthly M4 series over an exponential smoothing grid and field three rules ordered by machinery: a fixed cadence with no monitoring, a one-parameter tracking signal after Trigg and Brown on the deployed model's one-step errors, and an evidence-gated trigger that fits a twelve-candidate grid for a validation score gap. Model form matters little on average here, so a cheap cadence matches full updating. The ordering within that band is informative. The tracking signal matches full updating at the three-step benchmark, but that edge dies under rate matching: against a cadence making the same nine searches, the gap is null at every horizon. What survives is the detector, once separated from the selector. The score-gap signal loses to its cap-matched cadence when a fire deploys the validation winner, but beats it at all five horizons when a fire re-selects by AICc, never worse at any lead; the validation-winner selector trades the first lead for the largest long-horizon gains, beating the cadence at horizon 18 by 0.4 percent (t of -9.0). Whether a gate repays its compute reduces to a break-even weight the evaluation reports. Everything rests on realized out-of-sample loss; the in-sample analogue does not predict degradation.

Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein's unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein's lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein's lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.

The present study provides a closed-form characterisation of Nash equilibria in N-player binary games with unidirectional dependencies. While general network games are PPAD-complete, prior work has established that trees or paths admit polynomial-time solutions via dynamic programming. We provide a deterministic characterisation for the subclass of directed cycle graphical games, demonstrating that non-zero boundary incentives linearize the topology into a feed-forward propagation. Under this Robust Incentive Structure, resolution is achieved in O(N) time: strict dominance guarantees a unique equilibrium; in its absence, pure strategy equilibria are governed by the Parity Condition, while a unique fully mixed equilibrium is guaranteed via induced payoff indifference. For non-robust regimes, we deliver branching rules. The transition-matrix formulation evaluates the search tree size beforehand. This transparency enables the inverse design of target equilibria in circular networks, making explicit the mechanics that remain opaque in numerical solvers.

In the sciences, regression tasks often require predicting high-dimensional outputs from few training examples. Multi-output Gaussian processes excel in low-data regimes but typically struggle with high-dimensional outputs. Compress-then-predict pipelines such as PCA-GP (principal component analysis plus Gaussian process regression) handle high dimensionality, but rely on bases optimized for reconstruction rather than prediction. To address this gap, we propose a model that represents each output as a linear-Gaussian decoding of a low-dimensional latent state drawn from a Gaussian process prior. By analytically marginalizing the decoder weights, we couple compression and prediction in a single objective that scales to high-dimensional outputs. We refer to this model as Gaussian process latent factor regression (GPLFR). We demonstrate GPLFR by building the first spatially resolved emulator of global climate models for rocky exoplanets.

In various application domains, there is a certain `null cell', inside a multinomial setup, where observations are recorded for the other cells, but where one cannot count the number of occurrences for the null cell. I develop inference theory for assessing such unknown numbers, counting the uncounted, in situations where counts are available for the other cells, via parametric modelling. The methods are used to estimate the number of persons killed in Guatemala during the Genocidio guatemalteco years 1978–1995. There are three carefully curated lists of killed people, where the information can be mapped to a Venn diagram with $2^3=8$ cells. Summing over the seven observed cells, $R=\hbox{47,803}$ killed individuals can be identified, but how big is $N_{0,0,0}$, and hence $N=N_{0,0,0}+R$?

Given multiple data matrices, many problems in statistics and data science rely on estimating a common subspace that captures certain structure shared by all the data matrices. In this paper we investigate the statistical and computational limits for the common subspace model in which one observes a collection of symmetric low-rank matrices perturbed by noise, where each low-rank matrix shares the same common subspace. Our main results identify several regimes of the signal-to-noise ratio (SNR) such that estimation and inference are statistically or computationally optimal, and we refer to these regimes as weak SNR, moderate SNR, strong estimation SNR, and strong inference SNR. First, we propose an estimator based on projected gradient descent initialized via spectral sum of squares and show that it achieves the optimal $\sin\Theta$ error rate under strong estimation SNR. These results are complemented by both statistical and computational lower bounds identifying the weak and moderate estimation SNR regimes. Next, we turn to statistical inference for the $\sin\Theta$ distance itself, and we show that our estimator has an asymptotically Gaussian distribution in the strong inference SNR regime. Based on this limiting result we propose confidence intervals and show that they are adaptively minimax optimal in the strong inference SNR regime, where adaptivity is measured in terms of the SNR. Finally, we show that adaptive confidence intervals are information-theoretically impossible below the strong inference SNR regime. Consequently, our results unveil a novel phenomenon: despite the SNR being “above” the computational limit for estimation, adaptive statistical inference may still be information-theoretically impossible.

In many settings one is often interested in determining whether two networks share some joint structural connectivity patterns such as communities. However, while communities may be shared across networks, edge probabilities may differ significantly. Therefore, in this paper we consider testing a general null hypothesis that two networks have the same underlying subspace, which in particular includes the setting that communities are the same for either stochastic blockmodels or mixed-membership stochastic blockmodels (even if edge probabilities are different). We propose a test statistic based on the Frobenius norm of the difference of the leading subspace projection matrices, and we prove that our test statistic, after appropriate centering and scaling, converges in distribution to a Gaussian random variable as long as the average expected degree grows at least logarithmically in the number of vertices. We then provide estimators for the asymptotic mean and variance and show consistency under a stronger signal condition, and we give the local power of our test when the networks are sufficiently dense. Our theoretical results are based on a limit theorem for the projection difference of empirical and true eigenvectors which can also be viewed as the one-sample version of our test statistic, and this result may be of independent interest. We demonstrate our results through numerical simulations and an application to US Flight data.

Let $S$ be the set of unit norm linear classifiers $\theta \in \mathbb{R}^d$ which correctly classify every point of a labeled dataset $(X_i,y_i)_{i=1}^n$, $X_i \in \mathbb{R}^d$, $y_i \in \{-1,+1\}$, with a possibly negative margin $\kappa$ fixed in advance. Under two natural data-generating distributions of the $(X,y)$ pairs – a Gaussian mixture model and a logistic model with Gaussian features – and in the proportional regime $n/d \to \alpha$ with small enough $\alpha$, we establish a large deviation principle on the event that a point $\theta$ chosen uniformly at random from $S$ achieves a given generalization error, with high probability over the choice of the data. The associated large deviation rate function is deterministic and describes the proportion, at the exponential scale in $d$, of interpolating classifiers having a given desired performance. As a consequence, we establish the following concentration phenomenon: all but an exponentially small fraction of interpolating classifiers have approximately the same generalization performance given by the unique maximizer of this rate function. We numerically compare this maximizer to the performance of empirical risk minimization by gradient descent and to the performance of a natural linear program, both finding a point in $S$, and deduce that in the overparametrized regime of small $\alpha$, these efficient procedures outperform the vast majority of interpolators, pointing to their nontrivial benign overfitting in this setting.

In many oncology clinical trials where overall survival is a key endpoint, patients are permitted to switch from the control arm to the experimental treatment arm or other suitable therapies. Switching can occur for various reasons, including disease progression. This violates the causal guarantees of randomized treatment assignment, resulting in biased treatment effect estimates. Existing methods often require strong assumptions, complicated model specifications, or both. In this paper, we propose a general framework that incorporates external controls to account for treatment switching in randomized controlled trials. Leveraging the synthetic control method and balancing weights from observational causal inference, we propose several estimators that use multiple imputation and time-varying weights to adjust for treatment switching. We also discuss approaches to selecting the risk set of external controls to impute from. Through extensive simulation studies, we show that our proposed methods lead to meaningful statistical improvements relative to standard adjustment methods that utilize external controls in naive ways or those that do not utilize external controls at all. We then demonstrate the utility of our external control-based approaches with two phase III oncology trials.

Data-driven causal relationship identification is pertinent to advancing understanding of complex systems both within and beyond science. Bayesian networks offer a probabilistic method for modelling generic causal relationships via directed acyclic graphs (DAGs). However, typical techniques for constructing Bayesian networks rely on optimization, which can be ill-suited for learning causal relationships because the underlying data may admit multiple chains of causation. More data-faithful representations of causal relationships would provide frameworks for constructing multiple causal maps that are consistent with the variability that is inherent in underlying data. Here, we show that entropy-based inference generates atlases of plausible causal relationships that are consistent with underlying data. On simulated noisy data of 2- and 20-node linear structural equation models, we sample a maximum-entropy ensemble of graphs that allow us to quantify the inherent structural ambiguity in underlying causal relationships. Our method shows that "optimized" DAGs can contain causal artifacts are not consistent across equivalently accurate topologies.

This paper addresses model order selection under large-dimensional, correlated, non-Gaussian noise. Sources are assumed to be embedded in additive Complex Elliptically Symmetric (CES) noise with an unknown Toeplitz-structured scatter matrix. We propose a two-stage robust framework: (i) a noise-whitening step based on a Toeplitz-rectified $M$-estimator of the scatter matrix, and (ii) signal subspace rank inference via large-dimensional Random Matrix Theory (RMT). Almost sure consistency of the proposed estimators is established, together with explicit RMT eigenvalue upper bounds separating signal from noise components, in the regime where the observation dimension $m$ and the sample size $N$ grow proportionally. Three estimation branches are derived, based respectively on the sample covariance matrix (SCM), Maronna's $M$-estimator, and the distribution-free Tyler $M$-estimator for whitening. The methodology is validated on synthetic data, real hyperspectral images, EEG recordings, and financial data, with significant gains over AIC and unwhitened methods.

We formulate stationary-density-preserving nonreversible perturbations of Fokker–Planck dynamics as gauge fields that deform relaxation spectra while leaving the invariant state fixed. When detailed balance holds, a similarity transformation maps the reversible Fokker–Planck operator to a Witten-Laplacian-type supersymmetric Hamiltonian; nonreversible gauges then appear as non-Hermitian perturbations that preserve the zero mode but modify the excited spectrum. This operator viewpoint gives a common language for relaxation gaps, circulating probability currents, hypocoercive acceleration, and finite control costs. We represent admissible gauge currents by antisymmetric tensor fields and identify the detailed-balance-violating Ohzeki–Ichiki force as a constant symplectic example whose infinite-strength limit is Hamiltonian dynamics. The continuous-time spectral gap alone does not select a finite gauge strength, so we introduce a finite-time regularized objective and an actor–critic procedure for learning the gauge. An exactly solvable anisotropic Gaussian Ornstein–Uhlenbeck benchmark separates the spectral transition from the finite-time optimum and shows that the learned gauge recovers the Lyapunov-equation optimum. A double-well benchmark then illustrates the same constrained selection in a nonconvex metastable landscape. Stochastic gradient methods enter this framework as physically relevant Fokker–Planck systems: mini-batch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.

Concordance indices are widely popular metrics for assessing the ability of predictive survival models to discriminate underlying risk levels. However, these statistics have also been criticized for using only the rank orderings of the model's predicted risk scores and being insensitive to important model features, such as the addition of strong predictor variables into the model. In this paper, we address these limitations by developing smooth concordance metrics that model the underlying risk discrimination probabilities as continuous functions of the predicted risk score differences, where the shapes of these functions are estimated from the observed data. As a result, these smooth concordance metrics assess model performance across the entire range of possible risk score differences, allowing one to identify specific scenarios where the candidate model performs especially well or better than other models. Simulations show that the proposed smooth concordance metrics provide more detailed information about risk discrimination performance and are much more sensitive to the addition of meaningful predictors. We apply these methods to compare predictive survival models for cancer recurrence.

We give a complete classification of doubly totally-umbilical submanifolds in the probability simplex. The probability simplex is one of the most standard statistical manifolds, and information geometry initiated by S. Amari and H. Nagaoka studies the statistical submanifold theory of the probability simplex. On the other hand, H. Furuhata defined doubly totally-umbilical submanifolds in the geometry of statistical manifolds, inspired by the surface theory of Euclidean space.

Sharing the financial impact of rare adverse events across a group can soften extreme individual burdens, but any participant made worse off by the arrangement has reason to leave. A credible mechanism must therefore provide each agent with a trustworthy cap on their future obligation and should be deployed only if the aggregate harm across participants is bounded. We formalise this as the Certified Allocation Problem: from finite data and without distributional assumptions, find a redistribution rule, produce obligation caps for every participant, and verify that no participant is made materially worse off. We propose Conformal Risk Sharing, which solves this problem by pairing an interpretable sharing policy with split conformal calibration. The sharing intensity is tuned on training data, while held-out calibration data produces distribution-free per-agent guarantees (valid under exchangeability). Experiments on synthetic and real-world data, including precipitation and energy-cooperative data, confirm that the framework can substantially reduce extreme obligations for high-risk agents while controlling harm to others.