Papers

Although probabilistic statements are ubiquitous, foundational disagreements persist about their understanding, as exemplified by debates between Bayesians and frequentists; moreover, it is unclear when and why acting on them actually leads to desirable outcomes. Here, we argue that every probability is the output of a prediction method, that is, it depends on both a particular way of constructing abstractions and a way of transforming them into predictions. Through this, we provide a unifying perspective on supposedly different kinds of probabilities and show that even supposedly objective ones are model-dependent. We demonstrate that when a finite calibration criterion is met, one can anticipate the distribution of utilities for a given policy and inform successful decision-making on finite sets of events. Based on the notion of prediction methods, inductive arguments, and the probability calculus, we explain the feasibility of the calibration criterion in many settings. Overall, we develop a coherent perspective on probabilities and their use, connecting key intuitions behind other interpretations along the way.

This paper develops estimator-level asymptotic theory for maximum Diaconis-Ylvisaker prior penalised likelihood for logistic regression with a jointly fitted intercept and nonzero prior slope direction in the proportional-limit regime. For $\mathrm{N}(\mathbf{0}_p, p^{-1}\mathbf{I}_p)$ Gaussian covariates and $p/n\to\kappa\in(0,1)$, a conditional convex Gaussian min–max analysis yields almost-sure convergence of the fitted intercept and a pseudo-Lipschitz empirical law for the slope estimator. This estimator-level law gives asymptotic limits for out-of-sample scores, classification error, optimal thresholding and oracle calibration. It also yields oracle-adjusted fixed-block $Z$-statistics under isotropic Gaussian covariates, and yields the main ingredient in establishing the limiting distribution of the penalised likelihood-ratio test statistic and identifies the rescaling to recover the nominal chi-square distribution. We extend these results to Gaussian designs with arbitrary deterministic mean and positive-definite covariance via affine centering and whitening and discuss extensions to subgaussian covariates. Finally, we propose a consistent response-moment estimator of the oracle parameters entering the state equations that govern the slope limiting law and are required for feasible inference.

Muon replaces matrix momentum with an approximately orthogonal polar direction, but its geometry depends on the matrix representation. For convolution, standard unfolding describes a local patch map rather than the convolution operator. We introduce Muon-C, an operator-aligned optimizer that represents kernel momentum as frequency-wise channel-transfer matrices, polarizes these blocks independently, and uses a critical Fourier grid to return updates exactly to the original finite kernel support. We show that the new geometry arises from combining the block partition and Fourier coordinates. The exact-polar direction is a linear minimization oracle under the critically sampled convolution norm. Its worst-case guarantee relative to the continuous convolution-operator norm is never weaker than unfolding and is strictly stronger for $3\times3$ kernels. On CIFAR-10 flow matching with matched applied-update RMS, Muon-C reaches 9.87 FID at 40k iterations, compared with 22.26 for unfolded Muon and 51.31 for Adam. It reaches their final quality using $0.62\times$ and $0.64\times$ their model FLOPs, respectively. Under equal tuning budgets, Muon-C achieves 3.42 FID. Gains persist across data scales and transfer to classification across convolutional architectures.

Recovering modal amplitudes from a few sensors requires regularization, and the shape of the penalty decides how strongly each mode is shrunk. The best shape depends on the truncation noise that discarded modes add to every measurement, which is unobserved and could be room-specific. We identify a diagonal saturation principle: for a single snapshot, isotropic truncation noise makes the Bayes-optimal Tikhonov shape a closed-form power law Gamma_k proportional to lambda_k^|s|, set by the prior alone and independent of the domain. The off-diagonal anisotropy equals the Herfindahl index H of the discarded-mode power exactly, Weyl's eigenvalue counting law fixes the number of discarded modes and hence H, and Berry's random-wave conjecture enters only through a diagonal term D, which we measure directly. Together they predict an approximately flat loss landscape over the penalty exponent, which leaves a learned diagonal regularizer little to gain from tuning the exponent per room. On simulated acoustic rooms the closed form stays within 5.83% of a per-room power-law oracle across observation windows, and three learned diagonal architectures track its error, the closest within a fraction of a point and none more than 3pp above it, while learning qualitatively different spectra. In the truncation regime no tested method consistently improves on the closed form, including Learned Iterative Ridge, which couples modes across layers. The framework extends to heat diffusion through a known Green's-function correction with no new free parameters.

A popular class of causal inference methods addresses confounding through weighting, which reweights treated and control groups to balance their covariate distributions without using outcome information, thereby preserving a design-based perspective. In this paper, we propose sliced $L^p$ distributional balancing (SLDB), a family indexed by $p\in[1,\infty)$ that measures imbalance by averaging squared $L^p$ distances between the cumulative distribution functions of one-dimensional linear projections. The Cramér–Wold device ensures that this criterion identifies equality of multivariate distributions, while projection reduces its computation to sorting-based one-dimensional operations. Because our method lies outside the maximum mean discrepancy (MMD) framework underlying many existing distributional balancing methods, their theoretical and computational tools do not directly apply. We therefore develop a computationally efficient projected subgradient descent algorithm for estimating balancing weights, offering improved computational complexity over MMD-based methods. Furthermore, we establish a novel theoretical framework for SLDB-based causal effect estimation and prove, under suitable conditions, root-n consistency and asymptotic normality, with the asymptotic variance attaining the semiparametric efficiency bound. Finally, we develop inferential procedures that do not require augmentation with an outcome model, thereby retaining the design-based principle. Simulation studies and a real-world application demonstrate that SLDB performs competitively with existing methods.

Signal correlation and PnL correlation are correlations over different index sets - across assets at each date versus across dates for scalar payoffs - and practitioners often treat the first as a proxy for the second. We give an exact decomposition that shows what that proxy sees and what it discards. We recall that at each date a normalized signal's projection onto the realized demeaned return direction is its realized cross-sectional Pearson IC, so that fixed-signal PnL is return dispersion times IC (Qian and Hua, 2004), and we identify the normalized similarity of two signals in the orthogonal complement as their partial correlation controlling for realized returns; the residual rotational freedom is an orthogonal gauge whose invariants are the transverse Gram matrix. Signal correlation therefore equals an uncentered IC cross-moment plus transverse similarity, while Pearson PnL correlation centers and dispersion-weights the IC series alone. Our main result is a non-identifiability theorem: absent constraints on transverse geometry, neither correlation bounds or orders the other, which sharpens the simulation finding of Sorensen, Qian, Schoen, and Hua (2004) into an exact statement. For normalized ensembles the transverse Gram matrix re-enters through the normalization denominator; the ex post optimal combination under Euclidean and general covariance-risk metrics is the classical multiple-correlation bound, and we show that the two rules agree for every longitudinal exposure if and only if the risk metric is isotropic on the signal span. The classical exact null law of partial correlation under isotropy, a synthetic mechanism illustration at weak-IC scales, and inference guidance under temporal dependence complete the treatment.

The orthogonal factor model has been a very useful tool in uncovering covariance structures in a set of variables through a smaller set of underlying factors. This old model is suitable for continuous variables with unbounded support, since the most common assumption for the observables and the factors is multivariate normality. In this work, we propose a factor model for binary data. Factors are negative dependent, so they avoid each other. We study the theoretical properties of the model and carry out a full Bayesian inference. We illustrate the performance of our proposal with simulated and real data sets and compare with the traditional benchmark.

Time series foundation models (TSFMs) are increasingly pre-trained on synthetically generated time series trajectories, where the data generating process is known. Current pre-training recipes are based on loss objectives which compare TSFM outputs to realized future values of each trajectory. We instead propose loss objectives which compare TSFM outputs to the conditional forecast distribution of each trajectory, a procedure we call synthetic data distillation (SDD). SDD corresponds to a Rao-Blackwellization of the training objective, in that it leaves the expectation of stochastic gradients unchanged while provably reducing the covariance of the stochastic gradient under the Loewner partial ordering. We empirically validate SDD on a TSFM model family of sizes from $4$M to $2.5$B parameters, and observe faster convergence of validation loss at every model size: on Gaussian Process data, SDD attains or improves upon the Status Quo loss whilst requiring $10\%-40\%$ less training iterations.

Odor quality is commonly represented using high dimensional descriptor profiles, yet their low dimensional organization remains unclear. We investigated whether a two-dimensional hyperbolic embedding can provide an interpretable representation of this structure. We applied hyperbolic metric multidimensional scaling to two complementary datasets: 480 Sagar rating profiles from three participants rating 160 odorants on 15 continuous descriptors, and 4983 GoodScents–Leffingwell molecules annotated with 138 binary descriptors. The embeddings substantially preserved pairwise descriptor distances, supporting subsequent analyses of radial and angular organization. In Sagar, rating profile entropy was strongly and negatively associated with hyperbolic radius, with diffuse profiles closer to the center and concentrated profiles closer to the boundary. This radial organization emerged primarily at the level of the full descriptor profile, rather than any individual descriptor, and remained robust across alternative descriptor representations, participant specific analyses, and averaged ratings. Sweet, musky, fruity, pleasantness showed the strongest directional trends. In GoodScents–Leffingwell, active label entropy, reflecting descriptor multiplicity, increased with radius, whereas orthogonalized descriptor entropy, reflecting spread across orthogonal modes, decreased with radius. Related binary descriptors occupied coherent localized high-density regions. These findings reveal complementary radial and angular organization in the hyperbolic representation of olfactory descriptor data. They support hyperbolic mapping as an interpretable descriptive framework in which radius summarizes global profile properties, while the angular component captures continuous descriptor gradients and categorical organization.

Synthetic data has become a promising way to scale model training beyond limited human-generated data but it may also induce strong model collapse (Dohmatob et al., 2024), where any fixed fraction of synthetic data prevents model performance from improving under data scaling, leaving a non-vanishing excess risk floor. In this paper, we study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift. We establish finite-sample risk bounds for mixed and two-stage training, separating standard bias and variance from source-mismatch effects, namely fluctuation and persistent drift under mixing and filtered initialization bias under two-stage. These bounds reveal a sharp contrast: mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage, showing that collapse is not inevitable under a simple data curriculum. Under a random sketch model, we further obtain scaling laws for both protocols, with tight results for mixed training in the optimization-saturated regime. These laws show that larger models may amplify synthetic-induced degradation under mixing, and quantify how high-quality synthetic pretraining may reduce bias in two-stage training. Finally, we establish an exact finite-sample necessary-and-sufficient condition for two-stage training to strictly outperform real-only training under the same real-data budget and identical real-stage updates. Overall, our results highlight that synthetic data is neither inherently harmful nor beneficial; its effect depends critically on both its quality and the training protocol used to incorporate it.

Neural networks can leverage feature superposition to encode more concepts than dimensions, but cross-feature interference constrains the linear accessibility of simultaneously active features. By framing linear accessibility as a compressed sensing problem, we derive high-probability bounds for fixed supports under subgaussian noise, proving the sufficient dimension scales linearly ($d=O_{\varepsilon}(k \log m)$) rather than prior worst-case quadratic limits. We characterize the asymmetry between active and inactive interference and the trade-off between interference and observation-noise budgets. We then validate these bounds across system parameters through Gaussian-tail approximations. We also introduce IHT-SAE, which uses learned iterative refinement to improve feature recovery beyond the limits of linear availability. These results quantify the geometric constraints of the linear representation hypothesis, providing a framework for evaluating sparse autoencoders, compositional generalization, and neural interpretability.

We introduce this class of multivariate Kappa-Gaussian distributions by maximizing a type of Kaniadakis $\kappa$ entropy subject to certain first- and second-moment constraints. This differs slightly from the approach of Rényi and Tsallis, because the $\kappa$-entropy arises from a relativity-consistent deformation of the exponential and logarithmic functions and yields a family of distributions whose tails follow a power law, a family that actually reverts to the standard Gaussian distribution as $\kappa \to 0$. We then create a $k$-nearest-neighbor (NN) estimator for $\kappa$-entropy and describe it using the difference between two NN functionals involving sums of powers. Next, we establish $L^2$-consistency and asymptotic normality, using the well-known subadditive Euclidean functional and moment expansion framework for NN entropy estimators. Furthermore, using a scaling identity for the $\kappa$-exponential, we derive a closed-form formula for the entropy of the $\kappa$-Gaussian distribution itself. From this, we determine the exact threshold value $\kappa < 2/(m+2)$ at which the distribution preserves finite variance; this is analogous to the “degrees of freedom $>2$” condition in the Student's-$t$ distribution. With these components, we construct a non-parametric goodness-of-fit test and then verify it through simulations. Finally, we mention the Sharma–Mittal entropy, which combines the Rényi and Tsallis entropies as a natural direction for future work.

How much risk does a small reweighted training support retain? For finite weighted least squares with the minimum-norm learner, we prove the exact law $\Gamma_d(n)=3-n/d$ throughout $\lceil3d/2\rceil\leq n\leq2d-1$. The guarantee covers every observed feature rank and uses selections that preserve the full feature span. Balanced simplex anchors reduce dimension; positive-weight lifting and independent-line compression close the risk bound. Shifted coordinate pairs attain the matching lower bound. The complete dataset-level upper bound and sharpness construction are verified in Lean 4. At the smaller budget $(d,n)=(5,6)$, we also prove $\Gamma_5(6)=11/5$, matching the simplex-block prediction from $5=3+2$ over arbitrary interacting configurations. Circuit covers, comparison second moments, and circuit-plane probabilities give the sharp excess $6/5$, while polar-face geometry resolves shared rank-three circuits. The general simplex-block frontier connects these laws within the intermediate-budget selection problem.

Paraconsistent Annotated Logic with two values (PAL2v) annotates a proposition with a pair $(\mu,\lambda)$ of degrees of favorable and unfavorable evidence, drawn from a bilattice whose four extreme points are the values of Belnap and Dunn. In applications, however, the second coordinate is almost always taken to be the complement of the first, and several such annotations are then fused. We show that this practice determines the sign of the degree of contradiction in advance, independently of the data: for aggregation functions $F$ and $G$ the fused degree of contradiction equals $F-G^{d}$, where $G^{d}$ is the De Morgan dual of $G$, so that it vanishes identically when the two are dual, and has a constant sign otherwise. In particular the componentwise minimum used throughout the literature yields only consistent annotations, and the conjunction of annotated logic only exact ones; five of the twelve logical states, contradiction among them, are essentially unreachable by construction. We also give a decision procedure for the twelve states that is provably total, in place of the inequality lists in current use, which assign two states to about half of the non-extreme region. We then propose a construction in which the two coordinates are computed from disjoint families of evidence, show that every logical state is attainable. Educational assessment provides a natural instance, since summative and formative instruments are institutionally distinct: mapping examinations to $\mu$ and disengagement to $\lambda$ makes the degree of contradiction equal to the gap between what a student demonstrates and what that student invests. Simulations over three cohort profiles illustrate the classification.

Membership inference attacks (MIAs) have emerged as critical tools for auditing privacy risks in large language models (LLMs), aiming to determine whether a given text was included in a model's training corpus. However, most existing MIAs require access to per-token logits or probabilities, making them inapplicable in practice to proprietary LLMs that expose only textual continuations. To address this underexplored setting, we propose Word-level Probability MIA (WPMIA), a statistically principled MIA for strict black-box privacy auditing. WPMIA estimates word-level generation probabilities via Monte Carlo sampling with local kernel smoothing, then aggregates these estimates into a sequence-level likelihood estimator. Furthermore, WPMIA constructs the likelihood conditioned on different prefixes, thereby amplifying the distributional differences between members and non-members. We evaluate WPMIA across various open-source LLMs and find that it consistently outperforms existing black-box baselines. Importantly, we also evaluate WPMIA on modern proprietary LLMs, including GPT-5-Chat, Gemini-2.5-Flash, and Claude-4.5-Haiku, achieving an average TPR@5%FPR of 42.0 across these models. These results offer a sound foundation for future research on strict black-box membership inference. Code is available at \href{https://github.com/niusj03/WPMIA}{https://github.com/niusj03/WPMIA}.

Let $(X _n )_n$ be a sequence of i.i.d. $R^d$-valued random variables uniformly distributed on a compact subset $M$ of $R^d$ . In this work, we study the asymptotic behavior of maximal packings of this compact subset $M$. Under some assumptions on $M$ (in particular, that it is a smooth submanifold of $R^d$ without boundary), we establish a precise extreme value law (of Gumbel type) for the minimal distances from the sample $(X_1 , \cdots, X_ n)$ to the centers of maximal packings. This result leads to explicit asymptotic confidence bounds for this maximal packing and thus to the support $M$. A distinctive feature of our contribution is the explicit derivation of the scaling sequences in the Gumbel convergence, depending only on the geometry of the support M. These formulas provide interpretable confidence bounds for $M$, which represent a novel complement to previous approaches such as those by Fasy et al. (2014). Our approach bridges geometric and combinatorial probability arguments and relies on analytic tools such as the Lambert $W$ function. In addition, we provide examples of main sample spaces in directional and circular statistics (circle, sphere, torus), along with simulations that illustrate and support the theoretical findings.

Policy decisions often hinge on conventional p-value thresholds, which ignore economic costs and benefits of further data collection. This paper recasts statistical significance as a choice between making an immediate policy recommendation and deferring it until further evidence is collected. The welfare-optimal decision corresponds, under minimax regret, to a statistical test whose level depends on the cost and precision of additional evidence. Inverting this rule, we introduce and recommend reporting the abstention-value (A-value) to determine where additional data collection is most needed. The A-value defines the break-even welfare cost of abstaining and recommending further experimentation given the initial evidence. Computing the A-value only requires a point estimate and its standard error, imposes no prior assumptions on policy effects, and can be converted to a monetary research budget using inputs already frequently used by regulators and funding agencies. When experimentation capacity is limited, we show that prioritizing additional data collection where A-values are the largest yields strong finite-sample guarantees. Applications to anti-poverty programs and to a medical study illustrate the practical benefits of using A-values.

Wasserstein barycenters are a fundamental tool for the analysis of distribution-valued data, but their computation and statistical analysis become challenging in high-dimensional and multi-sample settings. The unregularized barycenter problem is computationally demanding and suffers severely from the curse of dimensionality, making regularization essential in many applications. While regularized optimal transport methods are widely used, approaches based on entropic penalization tend to overspread mass and may blur the geometric structure of the solution. Motivated by barycenter estimation, we propose a more general methodology based on sparse regularized multimarginal optimal transport (RMOT). Our formulation creates sparse transport plans and is designed to avoid this overspreading effect. We establish consistency, central limit theorems, and bootstrap validity for the optimal potentials and couplings for the empirical RMOT problem. Specializing to sparse regularized barycenters, our results yield a uniform weak limit and bootstrap consistency. Numerical simulations show that sparsity plays a fundamental role in preserving the structure of the solution and avoiding the overspreading behavior typical of entropic regularization.

Average treatment effect (ATE) estimation in observational studies is a fundamental statistical tool used frequently in social science, medicine, and other fields. These fields often work with sensitive data where privacy protections are important, so a differentially private mechanism for ATE estimation is highly desirable. Here we present two propensity score-based algorithms for ATE estimation on observational data, one improving the inverse probability weighting (IPW) method used in prior work, and the other using blocking on the propensity score (BPS). Both show lower error and less bias than prior work, with the BPS-based algorithm frequently reducing error by 75% or more compared to prior work.

We study the oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq \epsilon$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $\sigma^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(\sigma^2 \epsilon^{-3} + \epsilon^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with “sparse” noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

Statistical agencies increasingly rely on small area estimation to produce reliable estimates for subpopulations with limited sample sizes. These estimates are built from individual survey responses, so agencies must ensure that releasing them does not reveal information about any single respondent. We show that when a single draw from the posterior distribution of the Bayesian Fay-Herriot model is released, pure $\varepsilon$-differential privacy is unattainable, but the release satisfies formal privacy guarantees under Rényi differential privacy and zero-concentrated differential privacy without any noise being added, provided we treat the variance components as fixed. The key insight is that the posterior draw equals the posterior mean plus the Gaussian noise whose variance equals the posterior variance. The guarantee is thus governed by the sensitivity of the direct survey estimate and the posterior variance, and applies equally to a release of the posterior mean with that amount of noise added. For binary outcomes estimated with the Hájek estimator, the sensitivity equals the largest survey weight in the area divided by the sum of the weights. For the intercept-only model we derive exact coefficients describing how a change in one record propagates to every area's posterior mean, giving finite-sample per-area guarantees and a joint guarantee for releasing all areas at once that exceeds the largest per-area guarantee by at most a few percent in our applications. Two applications, poverty prevalence across 2,462 Public Use Microdata Areas in the American Community Survey and smoking prevalence across 52 substrata in the Washington state Behavioral Risk Factor Surveillance System, show that the guarantee is driven far more by the inequality of the survey weights than by the sample size, and that the shrinkage of the model tightens it substantially.

We study the projected power method (PPM) for synchronizing \(n\) unknown permutations of \(m\) objects under a possibly sparse uniform corruption model. Each pair is observed with probability \(p\), and an observed measurement is uncorrupted with probability \(\pi_0\) and is otherwise an independent uniform permutation. Under \(\log m=o(np\pi_0^2)\), we prove exact one-step recovery (with high probability) of each prescribed block for an independent estimate with a fixed positive majority of correct blocks. When \(np\ge C_0\log n\) and \(m=o(np\pi_0^2)\), we prove that one high-probability event yields a block-error contraction simultaneously for every estimate whose optimally aligned error is at most \(0.5-\epsilon\). The contraction factor is \(O(m/(np\pi_0^2))\) and the error floor is \(O(e^{-cnp\pi_0}+e^{-cnp\pi_0^2}+{\log n}/{n})\). Consequently, one update maps every possibly data-dependent estimate in this basin to vanishing block error, and all subsequent iterates remain almost exact uniformly over the iteration index. The one-step and trajectory results extend to independent, non-identically distributed, permutation-valued corruptions with mean \(m^{-1} \mathbf{1}\mathbf{1}^{\top}\). Under the uniform model, a reference-block spectral initializer has aligned block error \(O_{\mathbb P}(m/(np\pi_0^2))\), yielding an end-to-end almost-exact recovery guarantee. Under a stronger all-block signal condition, PPM reaches exact recovery after finitely many iterations. The theory transfers exactly to partial permutations with common support; for varying supports, we establish deterministic and probabilistic co-visibility margins.

We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for deterministic matrix-weighted sums of multivariate Poisson martingale differences generated by uniformly ergodic Markov chains. Under an $L^{(2+\eta)p}$-moment condition, $\eta>0$, we obtain the explicit bound $$O\left(p^3\|A\|_4^2+pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2\right),$$ where $A\in\mathbb{R}^n$ collects the moment sizes of the weighted increments. When these sizes are of order $n^{-1/2}$, the bound achieves the first optimal $O(n^{-1/2})$ rate for fixed $p$ and $d$. Consequently, we also obtain the first optimal $O(n^{-1/2})$ $W_p$ Gaussian approximation rate for multivariate additive functionals of uniformly ergodic Markov chains. Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence. First, building on the Ornstein–Uhlenbeck relative-score approach of Fang and Koike (2023), we formulate the bound in terms of antisymmetric Stein couplings while retaining the conditional tensor structure. Second, we develop a refresh-then-maximal coupling that combines an independent first-step resampling, which preserves the desired Stein identity, with a subsequent maximal coupling that provides effective control of the coupling increment. We apply these bounds to nonlinear Markovian stochastic approximation by decomposing normalized errors into matrix-weighted Poisson martingale sums and remainders. This yields the first sharp $W_p$ Gaussian approximation rates for both last iterates and Polyak–Ruppert averages. In particular, the optimized rates are $O(n^{-1/3})$ for polynomial-stepsize last iterates and $O(n^{-1/4})$ for Polyak–Ruppert averages, with an $O(n^{-1/2})$ rate for harmonic-stepsize last iterates under an additional spectral condition.

Normalizing flow Boltzmann generators retain a tractable pushforward density, but training with forward KL depends on target samples that may be biased or omit modes. As a result, a flow can miss target mass while its observed importance weights give a high effective sample size. We introduce the log-ratio variation $\X_\omega$, the mean absolute pairwise difference of the target-to-pushforward log-density ratio under a weighting measure $\omega$, and use it to define KLXX, a new loss function. Two log-ratio variations are added to the forward KL (denoted by the two X's): one weighted by the target to improve accuracy, the other by a mixture of quench and temper samples with pushforward samples to search candidate modes. We derive the Fisher–Rao gradient flow of KLXX, where both variations contribute nonpositive dissipation, and a fixed-surrogate error bound for KLXX. We use KLXX in an adaptive-staging Boltzmann generator, with importance reweighting at every stage. We bound the sampling error of its inference scheme when the stage weights are essentially bounded, and prove it asymptotically unbiased in the sample size. In the numerical tests, KLXX improves mode coverage over forward KL. It also improves the generator's per-stage diagnostics against the loss that built the schedule. The observables the generator recovers are close to independent references. The log-ratio variations thus supply information that the forward KL loss usually omits.

We introduce a flexible model for covariate-dependent multiple testing which can be encoded using a nonparametric Gaussian mixture model. Weight-localized predictive recursion (PRx), a new development in the methodology of Newton's predictive recursion algorithm, is then leveraged to estimate the components of this mixture model, allowing for recovery of the covariate-localized false discovery rate $\text{Pr}(H_i = 0|z_i,x_i)$ using a single, unified algorithm. This quantity represents the most direct extension of Efron's local false discovery rate to the covariate-dependent setting, and admits provable Bayesian FDR control properties under simple rejection rules. We introduce several procedures for estimating and thresholding the local false discovery rate, and show using various simulations and a real-data example that our procedures lead to increased power, tighter Bayesian FDR control, and more interpretable rejections. We furthermore show that this holds for fixed and randomized hypothesis labels, indicating that our proposed methods perform well under both frequentist and Bayesian interpretations of multiple testing.

We study sample covariance matrices $K = \frac{1}{N} \sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^* \in \mathbb{R}^{n \times n}$ in the proportional regime $n \asymp N$. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance $\mathbb{E} \mathbf{x}_i \mathbf{x}_i^* = \Sigma$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \mathbf{x}_i^* A \mathbf{x}_i - \mathrm{Tr} \Sigma A | \prec \| A \|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\Sigma$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta:= \mathrm{Im}\, z \geq N^{-1 + \tau}$, \[ \big| \langle \mathbf{u} , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)\Sigma \big)^{-1} \big) \mathbf{v} \> \big| \prec \sqrt{\frac{\mathrm{Im}\, \widetilde m_0 (z)}{N\eta}} + \frac{1}{N\eta} \] for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where $\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.

Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.

Weak Sparse Identification of Nonlinear Dynamics (WSINDy) provides a noise-robust approach for learning dynamical systems from data without requiring numerical differentiation. However, for high-dimensional systems, tensor-product libraries of candidate functions grow exponentially with the state dimension, making standard WSINDy expensive in both computation and memory. The Multidimensional Approximation of Nonlinear Dynamics (MANDy) addresses this scaling through a tensor-train (TT) representation of the candidate library, but does not provide a mechanism for sparse model selection. Here, we combine these approaches to develop TT-WSINDy, which performs the weak-form transformation, regression, and sparsification in TT format. We show that the TT formulation recovers the corresponding WSINDy regression problem and derive polynomial time and memory complexity bounds for the tensor-train sparsification procedure. Numerical experiments demonstrate robustness to measurement noise and computational savings for high-dimensional systems.

Receiver operating characteristic (ROC) analysis is commonly used to evaluate the diagnostic accuracy of continuous biomarkers. In practice, the true disease status may be unavailable and only a nominal disease status provided by an imperfect reference standard is observed. Existing nonparametric methods have been developed for ROC analysis in this setting, but may suffer from reduced estimation efficiency, numerical instability, or sensitivity to the choice of biomarker scale. We propose a semiparametric method based on a Box-Cox density ratio model, which links the biomarker distributions of the truly healthy and diseased populations while leaving the baseline distribution unspecified. A key feature of the proposed method is that the transformation parameter is estimated from the data rather than specified in advance, allowing the density-ratio structure to adapt to different transformation scales. We develop an empirical likelihood approach for estimation and an expectation-maximization algorithm for computation. We establish the asymptotic distributions of estimators of the ROC curve, area under the curve, Youden's index, and the sensitivity and specificity at the Youden-optimal cutoff, and develop bootstrap confidence intervals and a goodness-of-fit test. Simulation studies demonstrate that the proposed method provides accurate and numerically stable estimation and inference across a range of distributional settings without requiring the transformation scale to be specified in advance. The proposed method is illustrated using data from a malaria study.

This paper addresses the problem of estimating infectious disease prevalence under longitudinal testing programs that include scheduled, symptomatic, and contact-tracing testing. Our study is motivated by data from The Ohio State University, where a mandatory once-per-week COVID-19 testing and isolation program was implemented during the Fall 2020 semester, supplemented by additional testing for symptomatic individuals and identified contacts. In this setting, the probability of being tested depends on symptoms or contact-tracing status, creating a complex observation process. We develop a counterfactual framework that links the observation process to a hypothetical process in which infection is prevented. This formulation enables unbiased estimation of disease prevalence by modeling the testing process, possibly nonparametrically, without requiring explicit modeling of transmission dynamics, even though the testing and infection processes are jointly dependent.