Papers

The principle of Occam's razor, which instructs us to prefer simplicity in inductive inference, has attracted much scrutiny both in the philosophy of science and in machine learning. In either field, however, a justification for the principle has been elusive. In this paper, building on an earlier "core argument," I spell out a justification from statistical learning theory for the procedure of regularization: for trading off fit for simplicity. The means-ends argument is that in order to profit from theoretical reliability and "what-you-see-is-what-you-get" guarantees, one must implement a certain preference for simplicity over fit. This is a genuine methodological justification, which neither collapses to a purely pragmatic principle that we prefer simplicity for its own sake, nor to an ontological assumption that the truth is simple.

Generative Flow Networks (GFlowNets) have emerged as a flexible framework for amortised inference over discrete and mixed discrete-continuous objects, requiring only an unnormalised target density specified through a reward. In this work, we formulate forward-policy training in GFlowNets through the information geometry of the induced trajectory sampler. Treating the forward policy as an induced trajectory sampler, we show that its intrinsic first-order geometry is given by the Fisher-Rao metric of the trajectory family, and that the associated natural gradient provides the canonical local update whenever the corresponding Fisher information is computable or accurately approximable. We derive an exact decomposition of the trajectory Fisher into per-step conditional second moments, which clarifies when temporal score interactions vanish and when dense couplings remain under shared parameterisation. This leads to three computational regimes: settings with tractable exact Fisher information, settings where Monte Carlo estimators of the expected Fisher are sufficient, and structure-exploitable settings in which target locality or factorisation yields accurate approximations of the Fisher expectation. In the latter case, graphical-model tools such as exact marginalisation, separator methods, and belief propagation provide principled surrogates for natural-gradient updates. The resulting framework turns target structure into optimisation geometry and yields a tractable route to structure-aware forward-policy training in GFlowNets. We illustrate the framework empirically through examples comparing convergence and exploration behaviour under Riemannian and Euclidean optimisation.

We propose a resolution-adaptive Bayesian wavelet denoising method for noisy one-dimensional signals. Its central innovation is a spike-and-slab prior whose continuous slab mixes a compactly supported Wendland-type polynomial density with the more dispersed semicircle density. A low-dimensional empirical-Bayes trend produces data-adaptive mixture weights by resolution, while a data-adaptive support scale controls the common bounded interval. Thus, the method combines sparsity, explicit support control, and interpretable resolution-dependent shrinkage. Under squared-error loss, we derive the posterior-mean estimator and establish symmetry, boundedness, continuity, and limiting properties. We define fixed-hyperparameter bias, variance, and risk and develop an empirical-Bayes fitting procedure. Under a Laplace working likelihood, the Wendland contribution has finite-sum expressions, while the semicircle contribution is evaluated by stable one-dimensional integration. Simulations with the Bumps, Blocks, Doppler, and HeaviSine signals compare Gaussian- and Laplace-likelihood versions with universal thresholding, false-discovery-rate (FDR) thresholding, cross-validation (CV), Stein's unbiased risk estimate (SURE), the Bayesian adaptive multiresolution shrinker (BAMS), and a nonlocal-prior (NLP) method. In the primary Gaussian-error study, WS–Gaussian was the best non-NLP method in 24 of 36 cells, including 11 of 12 low-SNR cells, with a much more favorable computational profile than WS–Laplace. A real seismic acceleration record from the 2008 Chino Hills earthquake illustrates attenuation of rapid fluctuations and preservation of the dominant event. A semi-synthetic study using the processed trace as surrogate truth showed improvement over the noisy observation at lower and moderate SNRs, but not at the highest SNR.

Tensor cross-concentrated sampling (t-CCS) bridges entrywise sampling and t-CUR slice-wise sampling by observing entries only within selected horizontal and lateral slices. Existing t-CCS completion methods, however, assume that the observations are free of gross corruption. In this work, we study robust recovery of a third-order low-tubal-rank tensor from partial t-CCS observations contaminated by sparse, arbitrarily large outliers. We propose Robust Iterative t-CUR (R-ItCUR), a tensor-native algorithm that partitions the sampled tensor cross into two exterior blocks and an intersection block, applies adaptive blockwise Welsch correction for outlier suppression, and updates the low-rank component through projected blockwise gradient descent. By operating directly on the sampled cross, R-ItCUR avoids reconstructing the full tensor throughout the iterations, resulting in substantial memory and computational savings. Experiments on synthetic tensors, cardiac MRI data, and three-dimensional seismic data demonstrate accurate recovery and strong robustness to sparse gross corruptions. The results further highlight the importance of explicitly exploiting the cross-concentrated sampling structure in robust tensor completion.

Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing algorithms exhibit computational challenges: they either rely on preprocessing subroutines to enforce smoothness or on simulation-based training objectives, both of which can be expensive. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (AM). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random, smooth trajectories that agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and is devoid of expensive preprocessing. We provide ample numerical evidence suggesting that AM is competitive with or superior to existing algorithms on several benchmark problems from the existing literature.

Topological data analysis (TDA) uses topological techniques to extract meaningful shape-based information from complex datasets. Clustering is a central problem in data analysis, and there has been considerable recent interest in understanding how TDA can inform it. Existing approaches either cluster persistence diagrams directly under Wasserstein-type distances, which is computationally expensive, or use vector representations of diagrams. We propose a kernel $k$-means algorithm built on a convex combination of sliced Wasserstein (SW) kernels, one for each homology under consideration. Unlike other vector representations of persistence diagrams, the SW kernel is both stable and discriminative with respect to the $1$-Wasserstein distance. The method outperforms the baselines on two benchmark datasets and remains competitive on a third synthetic dataset, while being computationally efficient. It also outperforms both a single SW kernel on the union of all homology groups and an SW kernel computed directly on the point clouds. The convex combination assigns an interpretable weight to each $q$-homology kernel. We further validate that the weights identify the discriminating homology.

Anytime-valid inference enables analysts to continuously monitor their data and stop experiments early. However, the majority of these methods incur a certain conservativeness by remaining valid on infinite time horizons. In practice, a bound on the horizon may be imposed due to budgetary, practical, or ethical constraints. In this paper, we ask the question: "Is it possible to obtain sharper large-sample anytime-valid inference by forgoing validity beyond some finite time horizon?". We provide a positive answer to this question by proposing a family of statistical objects that we call "confidence horizons". These objects can be viewed as large-sample confidence sequences on bounded time horizons, or alternatively as group sequential repeated confidence intervals with a maximal number of interim peeking times. We make explicit connections to the group sequential boundaries of Pocock [1977], O'Brien–Fleming [1979], and Wang–Tsiatis [1987]. We derive closed-form distribution functions of certain statistics which can be used to calculate the asymptotic quantiles of confidence horizons exactly, sidestepping the repeated integration typically employed in group sequential methods. We illustrate the use of confidence horizons for treatment effect estimation in sequentially randomized experiments under adaptive Neyman allocation.

Empirical work often removes fixed effects, latent factors, or high-dimensional controls before estimating structural relationships. These transformations reduce confounding but may also remove identifying variation. We study linear panel IV after one equation-compatible nuisance projection under two-way dependence. The projected Jacobian determines which structural directions remain visible; the projected-score law determines their precision; and, on Gaussian fixed-rank strata, they combine in a Projected Information Matrix. We derive weak-identification limits with dimension-specific information accumulation, feasible factor-transfer conditions, identification-robust tests, bootstrap procedures for non-Gaussian interaction limits, and inference for the projected spectrum, rank, subspaces, and information matrix. Simulations show that a raw first-stage statistic above 500 can support the wrong sign while projected diagnostics reveal weak valid information. In an international monetary application, common projection substantially attenuates apparent foreign-output persistence, while Gaussian-reference Anderson–Rubin sets remain unbounded. Identification should therefore be assessed after nuisance removal.

Dimensionality reduction and effect estimation are central tasks in the analysis of high-dimensional data such as in genomics. Generalized bilinear models (GBMs) provide a versatile framework for these tasks, however, the statistical and computational efficiency of GBMs degrades rapidly as the number of sample covariates grows. To address this limitation, we introduce reduced-rank generalized bilinear models (RR-GBMs), which employ a reduced-rank sample coefficient matrix to model the effect of a large number of covariates without requiring an excessive number of parameters. In simulation studies, we find that when the true sample coefficient matrix is reduced rank or close to reduced rank, RR-GBM outperforms the standard full-rank GBM both statistically and computationally, providing more accurate estimates with lower computational burden. We develop a data thinning approach for model selection in the RR-GBM framework, facilitating rank selection. Furthermore, RR-GBM enables a new approach to visualizing the relationships among covariates and among features. We demonstrate the method in an application to Perturb-seq data for pancreatic cancer.

Safety surveillance increasingly involves repeated monitoring of many exposure-outcome signals in observational healthcare data, where sparse information, dependence across related signals, and systematic error can complicate inference. Existing frameworks typically focus on either correcting residual bias using negative controls or borrowing information across exposure-outcome pairs, but not both. We propose a multi-signal Bayesian sequential surveillance framework that integrates empirical bias correction with low-rank latent factor modeling. At each analysis time, a hierarchical Bayesian model learns exposure-specific bias distributions from negative control outcomes assumed to have null latent effects. Conditional on these distributions, low-rank latent factors are estimated across exposures and outcomes of interest to share information across correlated signals. As new data accrue, posterior inference is updated sequentially, yielding bias-corrected posterior summaries of effect sizes across multiple monitored signals. We illustrate the method in a postmarket vaccine safety surveillance study using a large US insurance claims database.

In this note, we introduce the notion of ranked spreadness, a strengthening of the usual spread condition in which the elements of each member can be ordered so that their one-coordinate marginals decay geometrically with their rank. This additional structure removes the dependence on the maximum set size in random-containment estimates. We prove width-free hitting and weighted-concentration theorems for ranked-spread set systems, together with an elementary kernel-extraction theorem showing that ranked spreadness arises naturally in arbitrary distributions on small sets. Our main application is to the simulation of nonadaptive property testers by sample-based testers. If a one-sided tester has average query complexity $d$ and rejects every far input with probability at least $\delta$, then, for every integer $c>d/\delta$, it admits a one-sided sample-based simulation with expected sample complexity $O_{d,\delta,|\Sigma|}\bigl(n^{1-1/c}\bigr)$. More generally, if positive inputs are rejected with probability at most $\gamma$ and far inputs with probability at least $\delta>\gamma$, the same conclusion holds for every $c>d/(\delta-\gamma)$. In particular, for constant-query nonadaptive testers we obtain an exponent $1-\Theta(1/q)$, matching, up to the dependence on the rejection gap, the exponent conjectured by Fischer, Lachish, and Vasudev.

Missing values present a common challenge in statistical modeling, so handling them properly is an important research direction. Among the various mechanisms that can generate missing values, the most common is the missing-at-random (MAR) mechanism, in which the probability of missingness depends only on observed data and not on unobserved data. This paper addresses the problem of estimating a multivariate linear regression model with multiple random covariates in the presence of MAR values in both the response and covariate spaces using a maximum likelihood (ML) framework. The proposed methodology models the joint distribution of responses and covariates through a conditional-marginal factorization of a multivariate Gaussian distribution. This formulation can be interpreted as a reparameterization of the multivariate normal distribution when the variables can be naturally partitioned into responses and covariates. Parameter estimation is performed using the expectation-maximization (EM) algorithm, which facilitates the imputation of missing values while preserving the distinct roles of responses and covariates. We extend this framework to the model-based clustering setting by considering a mixture of multivariate linear regressions with multiple random covariates. This extension enables soft clustering under incomplete data and accommodates MAR values in both the multivariate responses and covariates. Hence, it represents one of the most general model-based clustering solutions for regression data currently available in the literature. The effectiveness of the methodology is demonstrated through a simulation study, and the advantages of the proposed reparameterization are illustrated using the Automobile dataset, which contains missing values.

Many prediction problems seek to infer an unobserved, instance-specific parameter that governs the distribution of an observable response, even though the latent parameter is unavailable for both historical and future instances. We develop LatentCP, a prior-free conformal framework that constructs uncertainty sets for latent distributional parameters using only observed context–response pairs and a specified forward model. The method first constructs a conformal prediction set in the observable response space and then retains candidate latent parameters according to the probability their induced response distributions assign to that set. This inversion provides finite-sample marginal coverage without requiring latent calibration labels, a unique inverse mapping, or knowledge of the latent mixing distribution. Because latent-set efficiency depends nonmonotonically on the response-space miscoverage rate, we further introduce a multilevel procedure that aggregates normalized incompatibility scores across several response sets and selects the aggregation distribution using an independent tuning sample. Across synthetic experiments, LatentCP maintains nominal latent coverage under weak forward identification, observational nonidentifiability, and latent heterogeneity and multimodality, where empirical-Bayes, likelihood-based, and proxy-label conformal methods can substantially under-cover. On a California wildfire real dataset, it produces spatially adaptive uncertainty sets for latent fire intensity. Independent tuning improves efficiency, while multilevel aggregation provides additional gains when different response levels contain complementary information, without sacrificing validity.

This paper examines how political violence in West Africa is distributed between urban agglomerations and their surrounding areas using spatially disaggregated conflict and population data. Covering seventeen countries from 2012 to mid-2025, the analysis shows that political violence remains clearly tied to cities and their peripheries with the majority of violence occurring within and around urban agglomerations. Large cities record the highest number of incidents within their boundaries, while smaller towns, which are more frequently situated near borders, typically experience higher levels of violence outside their boundaries. However, urban-centric violence has also declined since 2012, indicating a progressive ruralization of conflict. These findings provide important regional context to the urbanization of conflict narrative by showing that West African insurgent movements use both urban and rural environments strategically. They also underscore the ongoing vulnerability of peripheral populations and the need for mobile and community-based security approaches outside major cities. Finally, they highlight the importance of strengthening border infrastructure, transport networks, and civic institutions as tools of both development and security.

The drift diffusion model (DDM) is a cornerstone of cognitive decision-making research. Although numerous estimation methods exist, researchers continue to seek inference approaches that are both fast and flexible across diverse study designs. Amortized Bayesian inference (ABI) can provide nearly instantaneous inference for complex stochastic models like the DDM, but neural networks trained for one study design cannot generalize to others. In this paper, we propose a divide-and-conquer framework that address this limitation. The core idea is that the DDM's independence assumption allows the full dataset to be decomposed into pairwise shards, each sharing a common structure that a single neural network can learn. Inference is performed on each shard separately and the resulting posteriors are combined via consensus MCMC to approximate the full posterior. Using simulated datasets, we evaluate the accuracy and uncertainty of this method. Our results show that the proposed divide-and-conquer approach achieves accuracy and uncertainty comparable to MCMC while reducing computational cost by several orders of magnitude. This work not only advances DDM estimation but also demonstrates a general strategy for improving the scalability and generalizability of ABI methods across diverse applications.

In mean-variance portfolio analysis, the efficient frontier represents the optimal trade-off between expected return and risk, assuming stable underlying parameters. This paper investigates portfolio fragility: the instability of optimal weights, risk-adjusted performance, and diversification when model inputs and construction choices are jointly perturbed. Combining constrained Markowitz optimization with variance-based global sensitivity analysis (Sobol indices), we map out how input uncertainty and portfolio-construction choices propagate along the target-return dimension. Using an empirical universe of multi-asset exchange-traded funds (ETFs), we find a distinct transition in the sensitivity structure: in the baseline experiment, lower target returns are dominated by l2 regularization, whereas aggressive return requirements become increasingly sen- sitive to the weight cap and expected-return perturbations. This shift coincides with a sharp drop in effective diversification and a rise in weight dispersion. We extend the analysis to a multi-universe fragility atlas, showing that under a com- mon absolute concentration rule, the smallest universe is weight-cap-driven in the aggressive return region, while larger sampled universes remain more often regularization-driven. The fragile frontier serves as a direct diagnostic tool to evaluate the structural robustness of constrained optimizers without altering the underlying allocation rule.

Self-consistency assumes the most frequent answer among sampled reasoning traces is the most reliable, but this can fail in causal reasoning: samples often repeat the same confounding error, and votes fragment across multiple valid answers, letting an invalid answer win despite a valid minority trace. We introduce CALVER (Causal Axiom-Level VERification), a training-free symbolic verifier that scores structured traces against Pearl's causal criteria, including -separation, backdoor adjustment, and intervention, and selects the highest-scoring candidate without consulting a reference answer. On CLEAR find-one-valid queries that admit multiple graph-valid answers, CALVER reaches 42.1% where plurality, a reward model, an LLM judge, and model confidence remain near 30% on identical frozen pools. Scaling the judge to 72B does not close the gap. In an audited clean-core subset, 11 of 21 graph-valid CALVER selections differ from the benchmark's listed answer while still satisfying the requested predicate. The advantage widens with the sampling budget and reproduces across ten published Bayesian networks, a second model family, and settings where the model must build the graph from text. CALVER also improves thresholded average-treatment-effect decisions against exact ground truth, generalizes to logic under a truth-table checker, and scores each candidate in milliseconds on CPU. CALVER needs only a causal structure, supplied outright or built from the text; wherever that holds, selection can aggregate via causal validity.

We develop a fully nonlinear structural vector autoregressive framework in which the contemporaneous structural mapping may be nonlinear and non-additive. Identification is achieved by exploiting variation in the conditional distributions of the mutually independent structural shocks induced by an observed exogenous variable. Specifically, a general contrastive learning framework that makes use of this variation together with the assumed exponential-family structure is employed to recover the shocks. Existing independent innovation analysis results identify such shocks only up to arbitrary componentwise invertible transformations, which is generally insufficient for structural econometric analysis. We strengthen this result by imposing a structured exponential-family specification for the conditional shock distributions. With the imposed sufficient statistics, the remaining ambiguity is reduced to a one-parameter transformed-scale map for each shock. We then show that, under a logistic specification used for the natural parameters, the identification is further strengthened up to permutation and componentwise sign changes. Once the shocks have been recovered, the fully nonlinear structural vector autoregression can be estimated using feed-forward neural networks, motivated by their universal approximation capabilities. The empirical application studies asymmetries in the responses of U.S. industrial production to the real oil price shock. We find modest asymmetries with respect to the sign of the shock and state of the economy. The accompanying R package iiasvar implements the introduced methods.

The performance of Support Vector Machines (SVMs) critically depends on the kernel function choice, which enables implicit mapping of data into high-dimensional feature spaces. While classical kernels like Radial Basis Function (RBF) remain popular, orthogonal polynomial kernels offer mathematically interpretable alternatives that can incorporate structured prior knowledge. This work extends the orthogonal polynomial kernel paradigm by introducing a novel family based on discrete $q$-Hermite I polynomials, a class of $q$-orthogonal polynomials that generalize classical Hermite polynomials through a deformation parameter $q$. We formally define the q-Hermite kernel and establish its validity under Mercer's theorem. The kernel's inherent boundedness properties naturally prevent annihilation and explosion effects without requiring explicit scaling mechanisms. Extensive experiments across 20 benchmark datasets demonstrate that the proposed kernel achieves competitive performance compared to both classical kernels and other orthogonal polynomial kernels, while offering advantages in numerical stability and computational simplicity. Our results confirm that $q$-orthogonal polynomials constitute a promising direction for kernel design, bridging mathematical elegance with practical machine learning applications, that provides conceptual and algorithmic resources that may be further extended to emerging quantum computing paradigms. To facilitate full reproducibility, we provide the complete implementation and experimental pipeline in an open-access GitHub repository at https://github.com/Kokechacho/SVMs-QSVMs.

We study score learning for reflected diffusion on bounded domains. Reflection keeps trajectories feasible but does not ensure that the learned score satisfies the boundary behavior implied by the forward process. With implicit score matching, integration by parts leaves a boundary term, and we show that it depends on one scalar at each boundary point: the diffusion- weighted normal component of the score, or conormal trace. The no-flux condition fixes this value while leaving the re- maining boundary components unrestricted; under anisotropic diffusion it generally differs from the ordinary normal score component. On hyperrectangles, our parametrization enforces the required trace without additional trainable parameters or a stochastic boundary estimator and, under regularity assump- tions, can represent the true score, whereas fixing an incorrect value creates an error that more data cannot remove. We ex- tend the construction to simplices and polygonal domains and identify reflection masking: hard reflection can keep samples feasible even when the learned trace is wrong, so post-reflection metrics may hide the error. Experiments show the clearest separation with less frequent reflection, anisotropic diffusion, and mass near intersections of constraints; under full reflection, final sample placement improves inconsistently, illustrating how hard repair can mask boundary-score errors and decouple score accuracy from downstream generation quality.

The effect size and nuisance parameters needed to appropriately size a clinical trial are generally not adequately understood before a trial begins. Through pre-planned early stopping rules, a conservatively planned group sequential design can effectively adapt the sample size for a clinical trial for a new treatment that is ineffective, very effective or minimally effective. One potential issue with such designs is that a substantial number of patients can be enrolled after a data cutoff for an interim analysis while data are being entered, cleaned, analyzed and discussed. A strategy of re-estimating the sample size at an interim analysis based on conditional power has been proposed to reduce somewhat this enrollment overrun issue. A purported advantage sometimes claimed is a smaller up-front planned sample size than a conservatively planned group sequential design. We demonstrate derivation of 2-stage group sequential and conditional power designs using the gsDesign R package and suggest comparing designs with comparable power using expected sample size calculations. While there are cases where conditional power designs may have small advantages, it is quite easy to derive very inefficient conditional power designs. This, along with the fact that a conditional power design may reveal something about the interim treatment effect, will often leave a group sequential design as the design of choice.

Scalable spatiotemporal modeling remains challenging because conventional methods rely on covariance models that tightly couple spatial representation and temporal inference, often leading to high computational costs. To address this difficulty, we developed coarse-to-fine spatiotemporal modeling (CF-STM), a framework that extends coarse-to-fine spatial modeling (CF-SM) to spatiotemporal settings. CF-STM represents latent spatial processes through multiscale locally weighted models, with temporal dependence modeled separately through state-space models defined at local centers. This covariance-free formulation achieves scalable computation while allowing temporal models to be flexibly specified without altering the spatial representation. Monte Carlo experiments demonstrate predictive performance comparable to that of alternative scalable space-time models at a substantially lower computational cost. An application to long-term residential land price data in the Tokyo metropolitan area shows that CF-STM flexibly captures complex spatiotemporal patterns while enabling interpretable inferences. CF-STM is implemented in an R package spCF (https://cran.r-project.org/web/packages/spCF/).

Operational evidence, i.e., evidence of operation without failure is an important component of confidence in the safety or reliability of a system in service, but it is costly to collect. When the context of operation changes, the relevance of previously collected operational evidence becomes unclear. This problem arises, for example, when an autonomous vehicle that has operated safely in one operational environment (the Target Operational Domain, TOD) is deployed in a different but related TOD. Existing practice often treats such evidence in an all-or-nothing manner: either it is fully reused, or it is discarded. Neither position is satisfactory when there are good reasons to believe that the new context is no worse than the previous one, but that belief is itself uncertain. This paper studies how to make post-change reliability claims by combining evidence from two TODs using Conservative Bayesian Inference (CBI), which combines the evidence from the new context with a weighted amount of previous context and partial prior knowledge through constraints on a set of admissible priors. This yields conservative posterior bounds on quantities of interest. A numerical example illustrates how pre-existing evidence from a previous TOD can be transferred, conservatively and transparently, to support reliability claims in the changed TOD.

Fractional Ornstein–Uhlenbeck (fOU) processes model temporal dependence and memory, including long-range dependence, while retaining the classical Ornstein–Uhlenbeck process as a special case. We extend the integral fractional Ornstein–Uhlenbeck (ifOU) process to a multidimensional setting for animal telemetry. Longitude, Latitude, and Altitude velocities are represented by coordinate-specific fOU processes driven by a multivariate fractional Brownian motion, allowing each coordinate to retain its own damping, scale, and Hurst parameters. We establish covariance validity, characterize the admissible cross-correlation region, and derive the asymptotic behavior of cross-covariances and separated increments. We develop procedures for finite-dimensional simulation, Gaussian likelihood inference, and conditional velocity reconstruction. Replicated simulations examine estimation of cross-coordinate correlations, while joint estimation of the complete parameter vector is illustrated for one three-dimensional trajectory. The proposed model is applied to telemetry records from five common noctule bats migrating in Germany, including three trajectories with Altitude measurements.

Latent growth models (LGMs) have been widely used to describe individual trajectories of change, average change patterns, and individual differences in change within a group. Structural equation model tree (SEM Tree) partitions samples through recursive partitioning using external covariates and characterizes between-group differences via parameters of prespecified SEM template models. An SEM Tree with an LGM as its template model may therefore serve as a useful classification method for longitudinal data. However, current SEM Tree implementations may face important practical obstacles, including high computational cost and frequent improper solutions caused by model misspecification or data characteristics (e.g., a limited number of measurement occasions). In the present study, we developed a matrix decomposition LGM tree (MDLGM Tree), which applies the matrix decomposition factor analysis (MDFA) approach, an eigenvalue-decomposition-based method for model estimation. Previous studies on MDFA and its extensions to SEM have shown that this approach can reduce the risk of improper solutions while yielding estimates highly similar to those obtained by maximum likelihood estimation. Through a series of simulations, we demonstrated that, compared with conventional SEM Tree, the MDLGM Tree greatly reduces the occurrence of improper solutions and is also highly efficient in terms of computation time.

Non-intrusive reduced-order models (NIROMs) have become a standard tool for approximating parametric partial differential equations from computer design of experiments while significantly reducing computational costs. However, assessing the reliability of their predictions remains a major challenge, particularly in extrapolation regimes or under limited training data. In this work, we introduce a framework for quantifying model-form uncertainty in NIROMs by combining a perturbative stochastic representation of reduced bases with distribution-free conformal-type methods. Starting from a deterministic reduced basis constructed from snapshot matrices, we model uncertainty through random perturbations defined on the Stiefel manifold, directed along the discarded modes, yielding stochastic reduced-order approximations whose induced variance reflects the basis-truncation error. A transport approximation gives a closed-form posterior variance that separates basis-induced from regression-induced uncertainty, without re-training the underlying Gaussian processes. We include this posterior variance within a conformal risk control calibration framework, that provides prediction sets with coordinate miscoverage guarantees. The calibration factor produced by this framework is itself an interpretable, scalar diagnostic of the quality of the uncertainty estimate. The methodology is evaluated on parametric PDE benchmarks and an industrial tire-manufacturing calendering process. Numerical experiments demonstrate reliable, locally informative uncertainty quantification that goes beyond the Gaussian predictive variance.

For a Markov kernel $T$ with an invariant probability measure $\pi$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^n_x$ and $\pi$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass $\mathrm{sing}(T^n_x \mid \pi)$ tends to $0$; and, 2.) asymptotic domination of the target: the singular mass $\mathrm{sing}(\pi \mid T^n_x)$ tends to $0$, as $n \to \infty$. Assuming a jointly measurable density for the absolutely continuous part of each iterate $T^n$ w.r.t. $\pi$, this criterion is sufficient and necessary for convergence. A positive minorant density version of it is verified in three cases: i.) $T$ has a positive transition density w.r.t. $\pi$; ii.) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis-Hastings algorithm; iii.) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In all results, the state space is a general measurable space with no structure beyond a $\sigma$-algebra. That joint measurability is assumed of the Markov kernel, not of the space; countable generation supplies it. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion.

We introduce the local quasi-linear (LQL) model, a differential equation-constrained local polynomial regression framework for the general first-order linear ordinary differential equation (ODE) $g'(x)=a(x)g(x)+b(x)$, extending prior work on differential equation-constrained local polynomial regression (DE-constrained LPR) for the exponential growth model. We derive closed-form DE-constrained local polynomial (DE1-$k$) estimators for arbitrary Taylor degree $k$, establish their asymptotic conditional bias and variance, and propose two approaches for estimating $a(x)$ and $b(x)$ when they are unknown. A simulation study across two structurally different ODEs shows that DE1-$k$ estimation with automatic degree selection reduces both estimation error and ODE-consistency error relative to unconstrained local linear regression. We then apply the framework to the firebrand burning-rate experiment of Albini (1979), modelling the density-loss curve of wind-driven firebrands with a physically motivated forced-convection ODE; the DE-constrained estimator outperforms local linear regression in the sparsest species-diameter groups, where physical structure is most valuable in compensating for scarce data. We further examine the robustness of the LQL model to misspecification against a local quasi-exponential alternative. Together, these results extend the DE-constrained regression paradigm to a broad class of physically motivated linear models and provide a practical estimation tool for fire science and other application areas where mechanistic knowledge is available but data are sparse.

We study some geometric consequences of the discrepancy between weak and total variation convergence of probability measures. We consider a sequence of probability measures on $\mathbb R^d$, admitting densities with respect to the Lebesgue measure, that converge weakly to a limiting measure but stay bounded away from it in total variation distance. We show that the sets on which the sequence passes from below to above the limiting density must grow unboundedly in perimeter, as measured by the $(d-1)$-dimensional Hausdorff measure. Moreover, this growth persists within a fixed compact set, so that it must reflect an increase in the geometric complexity of these sets rather than only an unbounded expansion in ambient space. We further provide a sufficient condition under which the number of connected components of the sets diverges, recovering a behavior that is closely reminiscent of the one-dimensional case, in which the number of oscillations of the sequence of densities around the limit grows without bound. We also show that this condition cannot be dispensed with in general, by means of an explicit sequence of measures in the plane whose passing sets remain connected in a single component at every stage while growing in length and complexity. Another sequence, built from cosine oscillations, illustrates the complementary behavior, in which the number of components diverges.

Existing low rank KV cache methods preserve either model weights or key variance, neither of which directly reflects the attention scores used during inference. We derive the expected attention score distortion caused by rank r key compression and show that it yields a covariance weighted low rank objective. Under a margin condition, controlling this distortion also improves top k recall. The optimal rank r solution has a closed form asymmetric factorization obtained from the SVD of the covariance weighted query key operator. This motivates SAKI, a training free KV cache index that directly preserves attention scores rather than key reconstruction quality. Across LLaMA 3.1 8B, Qwen 2.5 7B, Mistral 7B v0.1, and Llama 3.2 3B, SAKI outperforms key PCA at every tested rank. At rank 32, it removes 13 to 30 percent of PCA's remaining top 64 recall error, including improvements from 0.748 to 0.799 on LLaMA 3.1 8B and from 0.786 to 0.850 on Qwen 2.5 7B. It improves 68 to 89 percent of attention heads per model, with the largest gains in deeper layers. Predicted score MSE reductions closely match empirical measurements, with a Pearson correlation of 0.997, while ablation studies confirm that the gains arise from optimizing the attention score objective rather than covariance weighting alone. Analysis of the scoring operator further explains why weight only, invariant subspace, and key reconstruction methods can be suboptimal. SAKI uses random-matrix theory to separate genuine covariance signal from autocorrelated sampling noise, matching PCA with only 512 calibration tokens and adding value exactly where PCA sees no reliable signal.