Papers

We introduce the Intrinsic Hybrid Latent Diffusion Model (ILDM), a generative framework that integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds. While diffusion models (DMs) have achieved state-of-the-art results in high-dimensional data synthesis, they rely on large training datasets and ignore intrinsic geometric structure. Latent diffusion models (LDMs) address the high dimensionality by learning a latent space, but they typically impose a Euclidean structure, failing to capture the underlying manifold geometry, especially problematic in data-sparse regimes. ILDM addresses these limitations by interpreting the latent space as a chart of an unknown Riemannian manifold, with geometry and uncertainty quantified through a probabilistic decoder. The forward process is a hybrid diffusion that switches between Riemannian and Euclidean dynamics based on local uncertainty, where the Riemannian component is governed by a probabilistic metric tensor derived from the decoder. To learn the generative dynamics, we introduce an approximate denoising score matching method tailored to the hybrid diffusion setting, enabling a backward process defined by hybrid Langevin dynamics. Experiments on COIL-100, MNIST, and cardiac MRI datasets demonstrate that ILDM significantly improves generation quality, achieving lower FID and LPIPS scores compared to standard diffusion and latent diffusion models.

Federated adaptation of time-series foundation models (TSFMs) is attractive for building energy forecasting because meter data are private, distributed, and highly non-IID. However, a single parameter-sharing strategy is unlikely to serve all pretrained TSFMs or building clients: fully shared adapters can suppress building-specific temporal behavior, while fully local adaptation discards cross-building transfer. We propose a personalized federated sparse adaptation framework with a heterogeneous temporal mixture-of-experts (MoE) adapter placed after the pretrained TSFM representation. A sequence-level router maps each 168-hour context window to a top-$k$ subset of experts specialized for periodicity, long-range interactions, local variation, trend-residual structure, and multi-resolution behavior. We compare global FL, local training, and personalized FL variants with globally shared or client-private expert banks. Across 50 buildings and three TSFM backbones, personalization consistently outperforms Global FL-MoE and Local MoE, while the best sparse-adaptation strategy varies by backbone and metric. Routing behavior further reveals client-level expert specialization, expert concentration, and near-uniform routing across backbones, showing that federated TSFM adaptation should be both client-aware and backbone-aware.

Selective inference (SI) provides statistically valid $p$-values for hypotheses selected by applying an algorithm to the data, correcting for the bias that arises when the same data are used both to select and to test a hypothesis. Developing an SI procedure for a new algorithm, however, has required an expert to derive, and then implement, the selection event, i.e., the conditions under which the hypothesis is selected. Repeating this specialized effort for every new algorithm is why exact SI has so far been available for only a narrow class. We propose AutoSI, a framework that removes this barrier in two ways. First, AutoSI constructs the selection event automatically from the algorithm's individual operations, so the user only writes the algorithm as ordinary NumPy-like code and derives nothing by hand. Second, AutoSI broadens the class of selection events SI can handle: existing exact methods are limited to selection events characterized by linear or quadratic inequalities in the data, whereas AutoSI covers any algorithm expressible through rational functions of the data (ratios of polynomials). We prove that the $p$-values computed by AutoSI are exactly valid in finite samples. We demonstrate AutoSI on three feature-selection methods, each written in a few dozen lines of code. One of these methods, the lasso with its tuning parameter selected by cross-validated $R^2$, cannot be handled within existing exact SI frameworks and is made possible by AutoSI. Experiments on synthetic and real datasets show that the resulting $p$-values control the type I error rate (i.e., the false positive rate) at the nominal level while retaining high power.

An information theory-based framework is proposed in attempt to explain insistence on sameness in autism as an instance of a general behavior pattern in which an individual tries to reduce surprise and uncertainty. It offers a new definition of autism as an impairment in which cognitive functions are restricted to discrimination, memorization and prediction of tangible properties of the environment. An analogy between insistence on sameness and constrained minimization of the entropy metric is observed and examined for a set of assumptions that describe cognitive limitations of a person with autism. The metric is given by the formula $D_H(R, M) = H(R|M) + H(M|R)$, where $R$ represents sequences of random stimuli, $M$ is a memory that stores and retrieves them, and where $H(.|.)$ denotes their conditional entropies interpreted as surprise and uncertainty, respectively. It is first inferred that to minimize the metric an individual can learn about $R$ (and store that knowledge in $M$) or can restrict $R$ to the already known $M$. Then, it is concluded that insistence on sameness is a manifestation of the latter. Moreover, it is shown that the proposed framework: (1) Helps to quantify the concepts of surprise, uncertainty, sensory overload and deprivation, anxiety, comfort zone, disappointment, disorientation, pedantry, rigidness, observance or aberrant precision. (2) Leads to a list of guidelines for learning therapies and daily care routines, and allows them to be defined as optimization algorithms and implemented as programs for robotic live-in caregivers. (3) Can be validated with the help of a Turing test-like approach that requires no experiments involving individuals with autism. The framework-if positively validated-will provide formal foundations and design guidelines for therapies aimed at improving self-reliance of individuals with autism in basic activities of daily living.

Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong $L^2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier $0$ under projected restriction and $0.72$ under exact conditioning. A spatial regularity–cubature certificate closes the operator-realization term, a Bernstein argument gives a $\widetilde{O}(n^{-1})$ excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.

Filtering combines model predictions with measurements to estimate the probability density function (PDF) of a system state over time. The PDF often becomes highly asymmetric and even multimodal in nonlinear systems with oscillatory or chaotic dynamics. Such non-Gaussian features violate the single-Gaussian assumption underlying Kalman-type filters. To address this problem, Gaussian mixture filtering has been proposed. However, accurately propagating mixture components and adaptively adjusting their number and weights over time remain open challenges. Here, we develop an adaptive split-combine Gaussian mixture filter (AMF) that estimates the time evolution of asymmetric and multimodal PDFs by adaptively splitting and combining Gaussian particles without auxiliary online numerical optimization. Notably, the proposed splitting method guarantees a reduction in variance along a target level-set-point direction of a Gaussian particle. This enables accurate and efficient propagation of particles. We show that AMF consistently outperforms various baseline filters across diverse benchmarks, including single and coupled slow-fast Van der Pol oscillators and the Lorenz attractor. We also propose a parallel implementation of AMF, allowing high-fidelity PDF estimation with practical computational cost.

We study asynchronous optimization for finite-sum eigenspace computation in heterogeneous distributed systems. The theoretical foundations for asynchronous eigenspace computation remain scarce, with existing approaches offering limited coverage of dynamics directly on the Grassmannian under stale information. In this paper, we propose a Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients, retaining low per-update cost without global synchronization. The method employs an extrinsic polar update that preserves the intrinsic subspace geometry without requiring parallel transport of stale tangent vectors. Our analysis establishes a tight angle-dependent gradient-dominance characterization of the objective and a basin-invariance property for stale aggregated updates. These yield two-phase linear convergence, comprising an explicit broad-basin regime and a sharper local regime, with constants controlled by component spectral spreads. Experiments on serial and distributed PCA demonstrate improved sample efficiency and wall-clock convergence over representative baselines.

Gradient descent has been of particular interest in modern machine learning beyond sole focus on optimization. Implicit bias emerging from optimization, though not being encoded by the learning objective, often prevents from overfitting to spurious patterns. A typical instance is the max-margin implicit bias of a linear classifier, widely established for exponentially tailed loss functions. Even after having a given dataset separated, the parameter vector continues to evolve towards the max-margin direction asymptotically along the gradient descent dynamics. This phenomenon corroborates a frequent empirical observation of "train longer, generalize better." However, the max-margin convergence is an asymptotic phenomenon, and what is worse, this asymptotic convergence rate is significantly slower than pure convex optimization. Even so, the parameter vector along gradient descent dynamics commonly correlates with the max-margin direction positively (though not exactly) within considerably fewer iterations than the asymptotic rate. By shedding another light on this classical problem, this work aims to understand the mechanism of this early-stage alignment phenomenon. Our theoretical results demonstrate that the parameter vector weakly aligns with the max-margin direction within $O(\exp(\exp(-\delta)))$ iterations, where $\delta>0$ is the permissible alignment error, which is shown to be tight. By tracking the radial and tangential flows, our proof operates on the alignment dynamics directly with dataset geometry and gets rid of the asymptotic expansion, which is a key insight to establishing faster weak alignment.

Multimodal learning of images and tabular data is often impaired by ineffective representations, resulting in redundancy, dispersion, and generalization problems. To tackle this challenge, we introduce Graph-Enhanced Descriptor Sequencing (GEDS), a structured feature sequencing algorithm grounded in principles from the Column Permutation Problem (CPP). GEDS refines statistical descriptors of the features through similarity graph-based computations, systematically determining an effective feature sequencing. We incorporate GEDS within an order-aware efficient transformer framework, utilizing order-aware memory tokens that explicitly adhere to the derived feature sequencing via a dedicated loss function. Experimental results across multimodal benchmarks demonstrate that iStructTab effectively minimizes feature dispersion, improving predictive performance and robustness, and highlighting the significance of structured feature sequencing in multimodal learning.

Large language models are increasingly used for information seeking, yet semantically equivalent questions phrased in different ways can receive answers of considerably different quality. System prompts are widely employed to steer response behavior, but they are typically optimized for average-case quality, so some question phrasings may still receive incomplete or low-quality answers. To address this, we formulate a constrained mixed-strategy GroupDRO framework for system-prompt selection. Instead of optimizing the system-prompt text, the framework assigns weights to system prompts in an existing pool to minimize the worst-case information-quality loss across evaluation metrics and groups, while constraining the mean loss to stay close to that of average-based selection. Because pool generation and selection are decoupled, the method applies to any system-prompt pool and can leverage an ensemble of complementary system prompts rather than a single one. Across five LLMs on two bilingual medical and consumer-finance benchmarks, the constrained method reduces the Overall Mean, Worst 25% Mean, and Worst by 13.1%, 13.2%, and 13.7% on average relative to no mitigation while keeping overall quality close to Average selection. Its multi-prompt weights reveal complementarity across metric-group pairs. Code and data are available at https://github.com/Rainxu09/equitable-system-prompt-selection.

Recent proliferation of data-optimization integration has led to a range of methods that aim to improve the statistical performance of data-driven optimization decisions. However, while many of these methods are motivated intuitively from a robustness or regularization perspective, their resulting statistical benefits are often unclear and, even if available, are established on a case-by-case basis. We provide a systematic dissection of data-driven optimization formulations using the view of "directionally perturbed" empirical optimization (EO). Specifically, this umbrella of formulations, which we call "EO+", covers many existing data-driven optimization methods, including regularization, distributionally robust optimization, transfer learning, and analogous methods for contextual optimization. On the one hand, we argue that without additional, correctly specified, side information, any EO+ method can result in at most second-order improvements. This provides a negative conclusion, namely “no free lunch is possible", on the statistical power of EO+. On the other hand, we show that when leveraging side information that is geometrically effective, achieving first-order improvements is possible by choosing hyperparameters that are significantly larger than what is typically suggested in the literature. Moreover, we construct a principled methodology based on excess risk estimation, via either system knowledge or bootstrap resampling, to maximize the first-order gain. We demonstrate how this gain connects to the control-variate principle, a variance reduction technique in the Monte Carlo simulation literature, which helps explain why geometrically effective side information is necessary.

The Rashomon effect describes the phenomenon that many models can achieve nearly equivalent performance on the same learning task, with significant ramifications for robustness, feature importance, and customizability. These use cases motivate the computation of Rashomon sets: the set of all models whose regularized loss is near-optimal. Decision trees are one of the few model classes for which Rashomon sets can be fully enumerated, but this computation has always been conditional on a binarization of the original data, either restricting which splits each tree is allowed to make or substantially increasing the complexity of an already difficult combinatorial problem. We introduce the first algorithm that exactly enumerates decision-tree Rashomon sets while exploiting the ordered structure of continuous features. We further develop a relaxation for approximate enumeration and an anytime algorithm that progressively refines the set of candidate thresholds, producing increasingly detailed approximations that converge to the continuous-feature Rashomon set. Experiments show that coarse binarization can miss many trees, important features, and predictive multiplicity; our algorithms achieve orders-of-magnitude speedups over existing enumeration methods, with approximations providing further speedups while maintaining near-perfect recall.

We comment on the optimally-transported generalized method of moments (OTGMM) estimator proposed by Schennach & Starck (2026a) and give counterexamples to Theorems 2-6 under their stated assumptions. First, the assumptions used in the small-error analysis are insufficient for consistency in Theorem 2 and asymptotic normality in Theorem 3. Next, we consider the large-error analysis, in which Theorem 4 states that the OTGMM estimator is equivalent to a GMM estimator with modified moments. We show that in a scalar model, Theorem 4 selects a value that differs from the unique OTGMM minimizer and violates the OTGMM sample moment restriction. In an overidentified model satisfying the assumptions used in Theorems 5 and 6, the first component of the Lagrange multiplier has different probability limits under the OTGMM estimator and the GMM estimator with modified moments. Under misspecification, the population value selected by OTGMM depends on the transport metric and on which variables may be adjusted. We give a sufficient condition under which solutions of the modified moment equations also solve the original constrained problem at a given parameter value, and separate conditions for consistency and asymptotic normality of the OTGMM estimator. We also show that Assumption 16 does not imply the matrix bound used in the supplemental proofs and replace Assumption 16 with a matrix condition that yields the bound.

Estimating large and simulation-intensive discrete choice models (DCMs) requires repeated evaluation of utilities, probabilities, derivatives, and simulated likelihoods over many observations, alternatives, and draws. Existing DCM software provides mature econometric workflows, while recent GPU-oriented tools accelerate selected models, leaving a gap between econometric coverage and scalable differentiable computation. We introduce TorchDCM, an open Python package for discrete choice modeling that compiles choice data and model specifications into a unified PyTorch-native likelihood engine for estimation, inference, prediction, and structured reporting on CPU or CUDA devices. The package covers the principal econometric functionality available across Biogeme and Apollo, including multinomial, nested, mixed, ordered, latent-variable, and panel likelihoods. It also supports ragged choice sets, constrained parameters, covariance estimation, willingness-to-pay analysis, elasticities, and extensible likelihood components. We evaluate TorchDCM against seven other estimation packages in aligned synthetic and real-data full-estimation experiments. TorchDCM completes all 45 synthetic cases, runs fastest in every comparable synthetic case, and satisfies the prespecified final-log-likelihood tolerance in every comparison with at least two comparable solutions. More precisely, it reduces median runtime by 89.1%-99.7% relative to Biogeme and Apollo across model-data settings. CUDA provides an additional 12.0-71.0x speedup over single-core TorchDCM. These results establish a scalable and reproducible foundation for econometric estimation and differentiable choice-model development. The open-source package and executed examples are available at https://github.com/mbc96325/torchdcm.

Calibration requires a predictor to be unbiased after conditioning on its own predictions. Multicalibration asks for this guarantee simultaneously across a collection of groups. Many prediction tasks ask for several related features of the same conditional outcome distribution: variance is defined relative to the mean, skewness relative to both mean and variance, and conditional value at risk relative to a quantile. We study multicalibration for a sequence of $k$ properties in which each property is identifiable once the preceding properties are fixed. This framework includes Bayes pairs but does not require the properties to arise from a single loss. For every fixed $k\ge2$, we establish matching upper and lower sample-complexity bounds up to logarithmic factors under regularity conditions. Even with only polylogarithmically many binary groups, achieving multicalibration error $\varepsilon$ requires $\widetilde{\Omega}(\varepsilon^{-(k+2)})$ samples. Conversely, for any finite group family $\mathcal G$, we give a randomized learner using $O(\varepsilon^{-(k+2)}+\varepsilon^{-2}\log|\mathcal G|)$ samples. Thus the sample complexity is $\widetilde{\Theta}(\varepsilon^{-(k+2)})$ for polynomial-size group families. We instantiate the theory for three canonical examples.

Longitudinal prediction of Alzheimer's disease biomarkers increasingly informs clinical decisions, and a forecast is only useful if it also reports how much to trust it. Conformal prediction supplies this by wrapping any forecaster in a prediction band with a finite-sample coverage guarantee under exchangeability. However, standard population-level conformal prediction guarantees only marginal coverage and may mask substantial under-coverage within clinically important subgroups. We introduce a general mechanism-driven framework for auditing and repairing such subgroup under-coverage. Across two cohorts (ADNI, OASIS-3), two base forecasters, and nine attributes spanning genetic risk, demographics, and clinical severity, we find that population-level bands under-cover high-risk subgroups in 57 of 68 audited combinations, despite achieving nominal marginal coverage. We trace these failures to two mechanisms: (A) rarity, where a group-conditional band calibrated on only $n$ patients covers at most $k/(n+1)$; and (B) tail-heaviness, where a population-wide band is too narrow for a heavy-tailed subgroup and additional data cannot close the gap. Under-coverage falls disproportionately on patients with high genetic risk and disease severity (6.1 pp mean deficit, 95% CI [3.3, 8.9]), while demographic groups remain at the target level on average (0.0 pp, CI [$-1.9$, 1.7]). We pair each mechanism with a corresponding conformal correction: cross-conformal pooling for rarity, per-subgroup calibration for tail-heaviness, and a coverage-safe marginal floor when both arise. Together, these corrections restore target coverage for nearly every high-risk subgroup across both cohorts and forecasters.

We study the problem of aligning data from multiple modalities into a shared representation space, focusing on settings where strong pretrained unimodal encoders are available but cross-modal paired data are scarce. We propose a structure-preserving alignment framework, joint kernel entropic Gromov–Wasserstein Optimal Transport (JK-EGW), which maps multiple modalities into a common latent space by minimizing a quadratic optimal transport objective. JK-EGW leverages fine-grained similarity relationships within and across modalities to construct a global affinity kernel instead of relying on raw feature-space distances. Our framework naturally provides explicit control over the geometry and distribution of the latent embedding. On the theory side, we establish parametric sample complexity rate of $n^{-1/2}$, matching the corresponding rates for standard, entropic and Gromov–Wasserstein optimal transport. On the algorithmic side, we derive a scalable alternating procedure to solve JK-EGW with entropic optimal transport (EOT) updates through a low-rank kernel approximation and a variational lifting. This lifting scheme effectively relieves the burden of a quadratic objective, and allowing us to take the advantage of existing EOT solvers. Empirically, we focus on post-hoc alignment of embeddings from pretrained encoders in data-scarce regimes, and show that our proposed method achieves improved multimodal retrieval performance compared to existing alignment baselines.

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.

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.

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.

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.

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.

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 study contextual dynamic pricing with arbitrary covariate sequences and bounded, possibly nonbinary purchase quantities. Demand follows a semiparametric surplus-index model with an unknown linear valuation parameter and an unknown Hölder-smooth response. We impose neither concavity nor strong unimodality on revenue and allow nonunique optimal prices. We develop a pilot-corrected layered decision-partitioning policy that combines directional pilot estimation, local polynomial learning, predictable data assignment, and global action elimination. Pilot correction removes the first-order effect of valuation-parameter error, while permanent labels enable concentration under adaptive sampling. The policy attains the minimax smoothness-dependent horizon rate up to logarithmic factors; a matching lower bound already holds for a constant-context binary-demand subclass.