Papers

Boolean factor models, including prominent subfamilies such as Boolean matrix decompositions and cognitive diagnosis models, find broad applications ranging from social sciences to engineering. Despite their flexibility, a key challenge lies in establishing the identifiability of their graphical structures, which specify how latent variables influence observed variables. Existing identifiability conditions typically rely on the strong assumption of pure nodes, which may be unrealistic in many applications. We develop a novel approach leveraging the Hasse diagram to represent the distribution of observed variables and transform identifiability into a graph isomorphism challenge. Based on this, we establish sufficient and necessary graphical identifiability conditions that do not require pure nodes. We further derive equivalent algebraic conditions and develop an efficient Boolean satisfiability (SAT)-based verification algorithm. We extend the analysis to probabilistic Boolean factor models, establishing conditions for jointly identifying the graphical structure and additional model parameters without requiring pure nodes. Our results substantially broaden the class of identifiable and interpretable Boolean factor models by removing the pure-node requirement, yielding new theoretical insights, while also providing practitioners with a concrete and easily implementable tool to assess model identifiability.

Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution $P_{\mathrm{data}}$ from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance, for all $p\geq 1$. Specifically, given $n$ i.i.d. samples from $P_{\mathrm{data}}$, we show that, for every $d>d_p^\ast(P_{\mathrm{data}})$ and appropriately chosen network architectures and hyperparameters, the learned distribution $\widehat{P}^{\mathrm{FM}}$ satisfies $ \mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/\xi)\bigr)^{1/(2p)}$ with probability at least $1-\xi$, where $d_p^\ast(P_{\mathrm{data}})$ denotes the Wasserstein-$p$ dimension of the target measure. Our results demonstrate that flow matching naturally adapts to the intrinsic geometry of data and mitigates the curse of dimensionality, as the convergence exponent depends on the intrinsic rather than ambient dimension. These guarantees remain meaningful in high-dimensional regimes and provide a theoretical explanation for the empirical success of flow matching on structured data distributions under substantially more relaxed assumptions than those in existing analyses.

Covariance estimation is a key component of many applications in system identification and data-driven control. Although heavy-tailed distributions may lack a covariance matrix to estimate, the shape matrix provides a well-defined, scale-free generalization for the broad family of elliptical distributions. In this setting, practitioners often employ Tyler's M-estimator (TME), which is defined implicitly and is computed using a fixed-point iteration. The spatial sign covariance matrix estimator (SSCM) offers a much simpler alternative: it corresponds to one iteration of TME. Yet, the SSCM has been largely regarded as an inferior estimator due to its statistical inconsistency under fixed-dimensional asymptotics. By contrast, using second-order tools from random matrix theory, we establish that, under standard assumptions, SSCM asymptotically dominates TME in Frobenius risk when dimension and sample size grow proportionally. This advantage arises from implicit regularization, which reduces variance at the cost of a bias that vanishes as the dimension grows. Numerical experiments support the theoretical predictions even at moderate dimensions.

This paper studies the finite-sample bias of plug-in estimators for joint entropy, conditional entropy, and mutual information for finitely supported discrete random variables. Using the univariate reduction framework developed in our companion works, we derive explicit first-order bias formulas. For a pair (X,Y) with support sizes r and s, the biases are respectively -(rs-1)/(2n), -r(s-1)/(2n), and (r-1)(s-1)/(2n). We then propose bias-corrected estimators for the three measures and prove that the uncorrected and corrected versions are asymptotically equivalent, differing by a deterministic term of order O(1/n); consequently, they share the same asymptotic distribution. A simulation study validates the theoretical results, including a dedicated independence study and the relationship with Wilks' theorem. An application to a motor insurance portfolio, where X is the driver's age class and Y the claim severity class, illustrates the improvement brought by the bias correction for risk classification.

Differential privacy (DP) admits several notions, but the choice among them may affect both privacy analysis and utility. In this paper, we consider four mainstream curve-based privacy notions within a unified information-theoretic framework. For a fixed ordered pair of output distributions, we establish information equivalence among the two directional privacy profiles of $(\varepsilon,\delta)$-DP, the pair of hypothesis-testing trade-off functions, and the extended privacy-loss distribution. The exact Rényi differential privacy (RDP) curve joins this equivalence class whenever it is finite at some order greater than one. Under this mild condition, choosing among these notions changes only their semantic interpretation and computational requirements. In contrast, taking the maximum of the directional privacy profiles or compressing the RDP curve into a single zero-concentrated differential privacy (zCDP) parameter can lose information. We quantify the information loss between the exact RDP curve and its zCDP bound for standard noise mechanisms. This gap is zero for Gaussian noise but generally positive for Gaussian-mixture, Laplace, discrete Gaussian, and Poisson-subsampled Gaussian mechanisms. Moreover, this gap grows linearly with the number of independently composed mechanisms. Our information-theoretic perspective has practical consequences. At the same certified privacy level, retaining the full RDP curve rather than using zCDP reduces the required noise variance by up to $45\%$ for Gaussian-mixture noise in workloads comparable in size to the American Community Survey. For DP-SGD on Fashion-MNIST under Poisson subsampling, an RDP-based privacy accountant improves test accuracy by up to $8.73$ percentage points compared to a zCDP-based accountant when both are calibrated to the same $(\varepsilon,\delta)$ guarantee.

This study considers the problem of portfolio choice, where we recommend a portfolio to an investor to maximize the expected utility of their wealth. Our goal is to construct an asymptotically optimal portfolio choice rule in terms of expected utility regret, the difference between the expected utility of an oracle investor and that achieved by a portfolio chosen from data. We propose the Expected Utility Regret (EUR) rule, which jointly selects a portfolio class and estimates its weights. In a regular parametric return model, a single EUR rule attains both the minimax and the Bayes lower bounds, including their leading constants, without using the prior distribution that defines the Bayes criterion. We then derive the mean–variance and risk-parity portfolios as special cases of this framework. Under smooth increasing and concave utility, the EUR rule and the sample mean–variance portfolio attain the same leading expected regret when expected excess returns approach zero sufficiently fast. When the returns divided by their volatilities have a joint distribution that does not depend on the order of the assets, the EUR rule and the risk-parity portfolio coincide.

Many applications in statistics, economics, and physics require sampling from high-dimensional categorical distributions with local dependence structures. Examples include finite memory language models, Ising and Potts systems in statistical physics and protein folding, etc. In modern machine learning, discrete diffusions have emerged as a flexible approach for sampling such data, with strong empirical performance. Motivated by this, we develop learning methods with end-to-end sample complexity bounds for discrete diffusion with uniform noising under local dependence, which we model through low order Markov random fields (MRFs). Our main technical insight is a new pinning decomposition of the discrete score. It shows that unlike in continuous diffusions, the score decomposes into components where the dependence on time separates multiplicatively from the dependence on the target. Building on this decomposition, we propose a weight-sharing neural score learner and combine it with $\tau$-leaping to obtain an end-to-end sampling procedure. Rather than treating score-learning error as a black-box input, as is common in existing sampling analyses, we study the score learning error from finite data and derive optimal sampling guarantees with explicit dependence on the vocabulary size, the interaction order of the MRF, and the sample size. Moreover, our strategy trains a single score network across uniform noise levels while leaving the sampling discretization to be chosen at inference-time. This allows the same trained model to trade accuracy for computational cost as inference-time budgets vary. Numerical experiments on Potts, Ising, and tree-structured models show that weight-sharing score networks outperform fully connected ones for sampling long sequences.

The predictively oriented posterior offers principled uncertainty quantification, even under model misspecification. However, it does not admit an explicit density and therefore cannot be computed using classical Monte Carlo sampling algorithms. We remedy this by deriving an approximation to the predictively oriented posterior whose density can be evaluated point-wise, and whose approximation error decays rapidly. These results are illustrated on case studies from epidemiology, spatial statistics, and low-energy nuclear physics.

The error of regression using noisy function values naturally decomposes into an approximation error from the choice of model space and stochastic error from the noise on the observations. Statistical theory is mostly concerned with the stochastic error while approximation theory focuses on the approximation error, often subject to the assumption of noiseless observations. We combine the tools from approximation theory and statistics into a framework for bounding both errors in generalized least squares regression. The results cover weighted sup-norms and $L^2(\mu)$-norms, real- and complex-valued functions, and correlated noise under a range of tail and moment assumptions. Weighted Lebesgue functions and constants control the amplification of the best approximation error, while concentration inequalities provide bounds on the stochastic error. We organize these results into reusable bounds with explicit constants and establish sharpness where possible. For the common strategy of randomized regression we prove concentration of the Lebesgue function and constant around those of the corresponding $L^2(\mu)$-projection. For Fourier and Chebyshev regression on standard sampling grids, we obtain explicit, asymptotically tight $\mathcal O(\log m)$ bounds on the Lebesgue constants where $m$ is the model dimension. This improves upon the previous $\mathcal O(\sqrt{m})$ bound obtained for the regression setting and matches the bounds known for interpolation. A catalogue of approximation bounds and worked examples illustrates how to turn these results into explicit regression guarantees.

Model collapse arises when generative models are trained on synthetic data produced by earlier models. The phenomenon has attracted considerable attention because of its societal and technical implications. However, previous studies have reached seemingly contradictory conclusions: replacing real data with synthetic data causes collapse (Shumailov et al.), yet accumulating real data alongside synthetic data can prevent it. For diffusion models, we study an intermediate regime typical of finite-budget pipelines: all past datasets and the real data are kept, but each new model is trained on a fixed-size sample from this growing pool, so the real fraction vanishes without any data being removed. Experiments on a 2D spiral dataset as well as the image benchmarks (MNIST, Fashion-MNIST, and CIFAR-10) show that replacement protocol degrades dataset rapidly as in the literature, whereas the fixed budget degrades only partially, sparing some features. A linear-response model of the multi-generational parameter dynamics, analyzed by stochastic recursion, confirms that the two protocols differ: some features will be fragile and lost within a few generations for both protocols, while some will be robust and preserved over practically unbounded horizons under the fixed budget protocol.

Disentangling shared (joint) structures from source-specific (individual) variations is a fundamental task in multi-source data integration. Existing joint-individual models often rely on computationally intensive optimization or loose spectral bounds, leading to suboptimal separation accuracy and poor scalability. In this paper, we propose the Multi-Source Sequential Alignment Test (MSSAT). MSSAT leverages the geometric observation that true joint components manifest as closely aligned score subspaces across different data sources. By deriving the asymptotic null distribution of our alignment statistic, we develop a rigorous, resampling-free sequential testing procedure to accurately estimate the joint rank. Extensive simulations and real data applications, including a TCGA multi-omics dataset, demonstrate that MSSAT achieves superior separation accuracy and substantially faster computation compared to competing methods.

Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source–target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.

Gibbs samplers, and more generally component-wise Markov chain Monte Carlo algorithms such as Metropolis-within-Gibbs, can be implemented using either random-scan or deterministic-scan updates. How much convergence can depend on this choice of scanning rule has been a longstanding question. We study this problem through $L^2$ spectral gaps, measuring computational cost in units of component updates. For a $d$-block Gibbs sampler, the cost of random scan is at most twice that of any fixed deterministic scan, while the reverse cost ratio is at most of order $d^2$. We extend these comparisons to general reversible component-wise updates under the global block-wise contraction condition. If $K_j$ denotes the update of block $j$ and $P_j$ its Gibbs counterpart, and $\|K_j-P_j\|\leq \lambda_0<1$ for $j=1,\ldots,d$, then the cost of random scan is at most $2/(1-\lambda_0)$ times that of deterministic scan, while the reverse cost ratio is of order at most $d^2/(1-\lambda_0)$. Examples show that the cost bounds for random scan relative to deterministic scan are asymptotically sharp and that the joint dependence on $d$ and $(1-\lambda_0)^{-1}$ in the reverse comparison cannot be improved uniformly. This work was assisted by generative AI, including for formal verification of mathematical results in Lean. The human author reviewed and verified the mathematical content and takes full responsibility for the results.

The nonparametric maximum likelihood estimator (NPMLE) of a Gaussian location mixture maximizes the likelihood over the infinite-dimensional space of mixing distributions. The maximizing mixing distribution can be nonunique, and the classical bound on its number of atoms grows linearly with the sample size $n$. We show that a vanishingly small random perturbation of the likelihood yields exact polylogarithmic sparsity. The resulting randomly reweighted NPMLE maximizes a weighted likelihood whose independent weights, taken to be Gamma in our analysis, concentrate around one as $n$ grows. With high probability, it is unique, has $O\{(\log n/\log\log n)^d+\log n\}$ atoms in dimension $d$, nearly maximizes the ordinary likelihood, and estimates the mixture density at a Hellinger rate that is parametric up to logarithmic factors. This sparsity holds for the estimator itself, not for an approximation of it, and requires no support penalty. The proof rests on an effective-dimension principle for positive kernel mixtures: low-dimensional variation of the fitted values controls the support of every extreme point of the set of maximizers. Numerical illustrations verify that the reweighted NPMLE has Hellinger risk and support size comparable to those of the ordinary NPMLE.

Density peak clustering (DPC) connects each observation to its nearest neighbor of higher density and identifies cluster centers as high-density observations with unusually large nearest neighbor uphill shifts. The resulting uphill paths from observations to cluster centers, however, can be irregular and unstable in low-density regions, making the clustering assignments sensitive to local perturbations and obscuring the population geometry of the DPC graph. In this paper, we introduce gradient-guided density peak clustering (GGDPC), which performs a gradient ascent step before each nearest neighbor uphill search. We develop a stability theory that relates the GGDPC graph to the gradient ascent flow of the population density. In particular, we establish consistency of GGDPC under five complementary criteria: recovery of local modes, adjusted Rand index, dendrogram (cluster tree), path length, and waterfall measure. Together, these results provide new statistical, geometric, and topological interpretations of DPC-type clustering algorithms.

Iterative methods are extremely popular for solving linear ill-posed inverse problems. While deterministic convergence, or more precisely, semi-convergence of such methods is widely studied, the problem of quantifying uncertainty in the resulting reconstructions due to random noise in the observed data has received far less attention. In this work, we study uncertainty quantification for the Landweber iteration in linear inverse problems using the Radon transform as a motivating example. We interpret the Landweber reconstruction statistically by analyzing how uncertainty in the measured data is propagated through the reconstruction map. This perspective is closely related to generalized fiducial inference, where uncertainty about the parameter is induced by inverting the relation between the observed data and the unknown (fixed) quantity. We combine the resulting stochastic uncertainty with a theoretical bound on the regularization bias to construct confidence intervals for the reconstructed solution. The evaluation of the resulting uncertainty estimates require repeated operations with the forward operator that can become expensive in large-scale problems. To reduce this cost, we use randomized singular value decomposition to obtain a low-rank approximation of the normal operator associated with the Radon transform and incorporate this approximation in quantifying uncertainty. {{In particular, this requires that the additional randomization error introduced due to the use of such randomized techniques be included in our approach for uncertainty quantification.}} Numerical results show that the proposed method provides accurate reconstructions and reliable uncertainty estimates at substantially reduced computational cost.

We consider the augmentation of randomized experiments or trials with data from observational studies for the purpose of improving statistical precision. In particular, we focus on the Hybrid Augmented Inverse Probability Weighting estimator, designed to integrate predictions from several foundation models while preserving valid statistical inference. In this article, we extend the Augmented Inverse Probability Weighting framework to the estimation of the risk ratio, the quotient of the absolute risk of an exposed to an unexposed group. Our approach allows one to use information from black-box foundation models trained on external and possibly unstructured data, yielding an estimator of the risk ratio whose asymptotic variance is never larger than the one of the default estimator based on experimental data alone.

Consideration of frequentist type-I error or power based on certain fixed conditions may not fully embrace the aims of a Bayesian analysis; this is particularly true in settings where informative priors are involved. Calderazzo et al. consider adaptations of a testing procedure involving Bayesian borrowing from external control data. Particular caution may be necessary here in order not to condition on parameters and data simultaneously. We consider their example case and argue that (a) the conditioning scheme may have limited practical implications, and (b) if the main concern are assumption violations, these might be better addressed by adapting the likelihood function.

We establish non-asymptotic risk bounds for power Fréchet means in geodesic metric spaces with curvature bounded above. The observations form weighted, possibly infinite sequences of independent random variables whose laws and means may differ. We treat three settings. For $2$-Fréchet means in Hadamard spaces, we obtain a sharp mean squared error bound that becomes an identity in Hilbert spaces. For $2$-Fréchet means in CAT($\kappa$) spaces with $\kappa>0$, we establish variance and Wasserstein contraction inequalities with optimal constants depending on the circumradius $\varrho$ of the closed convex data domain, and obtain mean squared error bounds throughout the maximal range $\varrho<\pi/(2\sqrt{\kappa})$. For $\alpha$-Fréchet means in Hadamard spaces, $1<\alpha<2$, we derive finite $L^\alpha$ risk bounds under weighted $\alpha$-moment conditions, allowing even an infinite $\alpha$-moment of the population mixture. The proofs combine variance, quadruple, and contraction inequalities with a leave-one-out stability technique. Applications give prior and posterior bounds for Dirichlet-process Fréchet means and finite-sample guarantees for local constant Fréchet regression. The regression results require no response-space entropy condition and replace density smoothness assumptions common in earlier work with transport smoothness; for $\alpha<2$, the bounds remain finite even when the responses have infinite variance.

Parity superselection forbids direct measurement of odd Majorana observables. We learn time-independent, parity-covariant, $k$-mode-local Lindbladians on $m$ modes using even preparations and measurements with uninterrupted short-time evolution. Internal markers turn odd probes into even observables, and pair measurements among three separated markers calibrate their dynamical contributions. Signed Fierz inversion recovers canonical coefficients, while semidefinite fitting yields a valid generator. For finite-range models on known bounded-degree graphs with suitable marker access and a supplied weighted-strength bound $\bar\alpha$, entrywise error $\varepsilon$ is achieved using $\widetilde{\mathcal{O}}(\bar\alpha^2\varepsilon^{-2}\log(m/\delta))$ samples, without external modes or known nonzero coefficient locations. Recovery to diamond-norm error $\varepsilon$ costs an additional factor $m^2$, matching lower bounds for short-time experiments on fresh systems up to logarithmic and fixed geometric factors. Without a supplied graph, one idle ancillary mode per system mode and potentially nonlocal pair operations give total absolute coefficient error at most $\varepsilon$ per mode using $\widetilde{\mathcal{O}}_k(\bar\alpha^2\mathsf d^2m^{\lfloor k/2\rfloor}\varepsilon^{-2}\log(m/\delta))$ samples, under a supplied approximate coefficient-degree bound $\mathsf d$ and controlled weak-coefficient tails. All guarantees hold with probability at least $1-\delta$. For geometric models, fixed-support even observables can be predicted with logarithmic system-size sample complexity at fixed time and accuracy. We also give finite-volume and exponential-tail extensions.

Stochastic approximation provides a general framework for online estimation and optimization. Statistical inference based on the resulting estimates requires understanding their fluctuations around the target. For Polyak–Ruppert averaging, functional central limit theorems describe the normalized cumulative estimation errors through a Brownian limit. However, weak convergence alone does not determine how quickly the distribution of this process approaches that limit. We establish a quantitative functional central limit theorem for nonlinear stochastic approximation with polynomially decreasing step sizes and martingale-difference noise. Under suitable regularity and moment conditions, we derive an explicit finite-sample bound on the Prokhorov distance between each fixed scalar projection of the normalized partial-sum process and its Brownian limit. The bound reveals a step-size tradeoff: faster step-size decay reduces the contribution of the nonlinear remainder but increases that of the smoothing induced by the recursion. To assess the sharpness of the upper bound, we develop four lower-bound constructions, each designed to isolate one source of error. They show that covariance stabilization, finite-moment noise, algorithmic smoothing, and nonlinearity can each separately limit the Brownian approximation rate of the SA path.

Classical linear regression assumes that the relevant geometry of the predictor space is Euclidean and that all centered observations contribute to the least-squares fit in the same geometric scale. This paper proposes Riemannian Regression, a regression framework in which the usual vector differences are replaced by locally weighted differences induced by a data-dependent similarity structure. We introduce a generalized framework, termed Riemannian Regression, extending classic regression to any data endowed with a local distance structure. By equipping data tables with local metrics, we adapt regression model to incorporate manifold geometry. Given a similarity matrix $S=(S_{ij})$, obtained from UMAP, ISOMAP, or DBSCAN \cite{mcinnes,isomap,dbscan}, we define the dissimilarity coefficient $\rho_{ij}=1-S_{ij}$ and the induced subtraction $ x_i\ominus x_j=\rho_{ij}(x_i-x_j). $ A Riemannian center $g=x_\lambda$ is selected as a discrete Fréchet mean, and regression is performed on the Riemannian-centered variables $X_R=W X_{c,\lambda}$ and $y_R=W y_{c,\lambda}$, where $W=\operatorname{diag}(\rho_{1\lambda},\ldots,\rho_{n\lambda})$. The resulting estimator has the weighted least-squares form $ \widehat\beta_R=(X_{c,\lambda}^{t}W^2X_{c,\lambda})^{-1}X_{c,\lambda}^{t}W^2y_{c,\lambda}. $ The proposed approach preserves the linear form of the regression model while changing the geometry of the fit. The paper develops three ways to construct the local metric: UMAP-based fuzzy similarities, ISOMAP-based normalized geodesic distances, and DBSCAN-based density similarities. Simulated examples and the Abalone data set illustrate how Riemannian Regression can reduce the influence of locally anomalous observations and adapt to regions with different local densities.

Sampling several answers and keeping the one a verifier scores highest is one of the simplest ways to buy accuracy at test time. Its effect is reported as a scaling curve: accuracy against the number $k$ of sampled answers. The curve is cheap to draw and expensive to trust. A budget read off it is chosen after looking at every point, so only a band that covers all budgets at once protects the choice, and on a 100-question benchmark a fixed exact-binomial design needs 192,000 generated answers to certify 64 budgets to within $\pm1/32$ at 95%. Most of that cost pays for the wrong uncertainty. A benchmark is a fixed list of questions; at budget 64, about three quarters of the variance of a selected answer's correctness lies between questions, and an audit that revisits every question need not pay for it. We derive the minimax cost of certifying the whole curve, up to logarithmic factors. It has three parts: calibrating the tail of the score distribution, telling the questions apart, and within-question noise summed along the curve. At a single benchmark the last part sharpens to the variance of one answer's influence under the best allocation of answers to questions, which every valid audit pays and an audit that learns the allocation attains, up to a logarithm, as the precision grows. A paired audit built on an exponential inequality for two independent draws at the same question needs no pilot. On 185 held-out score pools it uses 0.74 times the answers of the cheapest competing certified audit at 64 budgets and 0.53 times at 1,024, and on a newly generated MMLU-Pro study it certified the curve with 79,133 answers, within 0.6% of what a cost law fitted beforehand predicted from the study's within-question variance. The same paths certify pass@$k$ and majority voting, and the bands extend to populations of questions and to answers that depend on earlier ones.

In many statistical settings, the available data and maintained assumptions do not suffice to uniquely identify the model parameters of interest. In such cases, one can only identify sets which are guaranteed to contain the true parameters. These are often characterized through linear programs that optimize over models compatible with the observed data. These programs can be infinite-dimensional in the optimizer and the number of constraints. We provide a unified way to characterize and solve such optimization problems by phrasing them as optimal transport problems on path spaces. This allows us to regularize the problem with an entropy penalty, recasting it as a multi-marginal entropic optimal transport problem, which can be solved efficiently via Sinkhorn iterations. In addition, it allows us to establish convergence of the regularized value to the sharpest bound, derive consistency rates for a plug-in estimator, and obtain asymptotic distribution for approximate bounds. The method is general and accommodates settings ranging from instrumental variable models with continuous variables to welfare estimation in heterogeneous demand models. We verify the statistical and computational properties in simulations and provide an application to demand estimation.

We study the robustness of the $F$-test in random design linear models, and reach a somewhat nuanced conclusion. On the positive side, one of our main results is that the size of the test is close to its nominal level as soon as either the distribution of the normalised error vector is close to uniform on the unit sphere, or the design matrix, after applying a suitable column space-preserving orthogonalisation scheme, is close to being uniformly distributed. This provides a sense in which the $F$-test is doubly robust. Our conclusion is reached by establishing a Kolmogorov to Wasserstein distance Hölder continuity property controlling the departure of the $F$-statistic from its notional $F$-distribution under the null. Writing $n$, $p$ and $p_0$ for the sample size and the dimensions of the full and null models respectively, we prove that the Hölder exponent is $1/3$ when $\min(p-p_0,n-p) = 1$ and $1/2$ when $\min(p-p_0,n-p) \geq 2$. On the other hand, these exponents are relatively small and cannot be improved in general, revealing that the size of the test may depart from its nominal level quite quickly as we move away from settings where the test is exact. In some cases, our conclusions may be improved by working with a local Kolmogorov distance that focuses on discrepancies between distribution functions in the right tail.

We study large deviations for rare clusters in sparse systems of moving particles. In the regime \(nr_n^d\to0\) and \(\rho_{k,n}=n^kr_n^{d(k-1)}\to\infty\), we prove a large deviation principle for the empirical measure of isolated \(k\)-particle trajectory clusters. The speed is \(\rho_{k,n}\), and the rate function is the relative entropy \(h(\,\cdot\,\mid\tau_k)\), where the finite reference measure \(\tau_k\) explicitly incorporates the underlying path law. As consequences, we derive free-energy variational formulas for bounded interactions and a hard-core constraint and identify the corresponding minimizing cluster law. The normalized optimizer provides the basis for Metropolis–Hastings sampling of interacting trajectory clusters. As an application motivated by chain formation and swarming in active particle systems, we calibrate the resulting stochastic model to experimental trajectories of magnetic micromotors and find that the fitted velocity scale varies systematically with particle size and magnetic forcing.

For estimating the mean vector of a $p$-variate normal distribution ($p \ge 3$) under quadratic loss, the positive-part James–Stein estimator dominates the original James–Stein estimator, but it possesses a non-smooth thresholding boundary. In this paper, by truncating the infinite series representation of the positive-part function to a finite sum of degree $m$, we propose a new class of smooth shrinkage estimators. We prove that the proposed estimator dominates the James–Stein estimator for any dimension $p \ge 3$, provided the truncation degree satisfies $m \ge 0.95\sqrt{p-2}$.

We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target $q_0 \propto w\,p_0$ at inference time. Relying on the stochastic optimal control approach, we observe that the exact drift correction is the gradient of the logarithm of Doob's $h$-function, and we study the problem of estimating it from a sample. In the present paper, we assume that the score of the pretrained model is available, that the tilting weight is bounded and positive, and that the reference distribution has a bounded support, no smoothness of the weight is required. Introducing a penalized least-squares risk in which the penalty is the residual of the space-time harmonicity equation satisfied by the $h$-function, measured in a dual Sobolev norm, we derive high-probability bounds on the squared error of the resulting guidance estimate. Since the penalty vanishes at the target, the estimator is free of regularization bias, and in favourable scenarios its rate of convergence is faster than the minimax rate of estimating first-order derivatives of a smooth regression function. Assuming that $w$ is bounded and positive with $\mathbb{E}_{p_0}[w^{-\mathrm{s}}] < \infty$ for some $\mathrm{s} \in (0,\infty]$, and that the reference data are compactly supported, we prove that the guidance is estimable in squared $L^2$ at rate $\varepsilon_n^{\mathrm{s}/(\mathrm{s}+4)}$, where $\varepsilon_n = n^{-2(\beta-1)/(2(\beta-1)+d)}.$ We also transfer the obtained bounds to the total variation distance between the marginals of the estimated and the exactly guided samplers, and illustrate the performance of the suggested approach with numerical experiments.

Despite the widespread use of Low-Rank Adaptation (LoRA), little is known about its dynamics in continual learning and the mechanisms by which low-rank updates affect catastrophic forgetting. We provide an asymptotically exact dynamical characterization of LoRA in a solvable two-task teacher-student model. In the high-dimensional online-learning limit, we derive a closed system of ordinary differential equations for a finite set of macroscopic order parameters, yielding exact expressions for the generalization errors throughout both the initial Task 1 learning phase and the subsequent LoRA fine-tuning on Task 2. The theory quantitatively matches finite-dimensional simulations and exposes two characteristic effects of LoRA: low-rank adaptation reduces interference with features learned on the first task, but its initialization slows adaptation to the second task. Building on this mechanistic picture, we analyze a state-dependent masking strategy that freezes hidden units carrying the strongest first-task representations and restricts adaptation to the complementary subspace. This structural partitioning markedly reduces forgetting, while preserving plasticity on the new task. Our framework further clarifies the role of adapter rank: transfer improves only up to the intrinsic dimensionality of the target task and saturates beyond it, while forgetting continues to grow with rank. These results provide a dynamical and geometric account of how low-rank adaptation organizes information across sequential tasks and are qualitatively reproduced on a sequential MNIST benchmark.

We develop a general second-order asymptotic framework for estimators constructed from invariant probability measures of data-dependent Markov processes. The framework yields a universal bias correction for smooth functionals, computed from the same invariant measure used to construct the estimator, without the resampling and repeated estimation involved in bias-correction methods such as the bootstrap and jackknife. The statistical input is represented by a random one-form, allowing differentials of log-likelihoods and general random criteria, as well as estimating functions, to be treated in a common framework. Under local regularity and localization conditions, we derive second-order stochastic expansions for averages with respect to these invariant measures and for decision rules defined through general loss functions. The expansions separate the contributions of the statistical input, temperature, drift, and loss, using precontrast geometry and an O-derivative induced by the loss. The framework includes maximum likelihood and posterior-based estimators and recovers existing bias-reduction methods based on the choice of prior or adjustments to estimating equations. In the likelihood setting, for the Jeffreys posterior $\mu_n$ and its Fisher–Rao Fréchet mean $\widehat z_n$, the corrected estimator $2\gamma(\widehat z_n)-\mu_n(\gamma)$ has frequentist bias $o(a_n^2)$ for every fixed smooth functional $\gamma$, where $a_n$ denotes the estimation rate. For coordinate squared loss, the theory yields a diffusion determined only by the likelihood and Fisher information. The mean of its invariant probability measure estimates the parameter with bias $o(a_n^2)$. This construction remains available even when no bias-reducing prior exists. Numerical results in the gamma shape–scale model illustrate this bias reduction.