Papers

Lacey used sparse domination to study the sharp weighted norm estimate of the maximal function of predictable multipliers in discrete time filtration spaces. Domelevo, Petermichl, and Škreb developed the self similarity argument known as sparse domination in an abstract martingale setting with a continuous time parameter. In our investigation, we establish sparse domination for discrete-time martingale transforms, introducing the novel concept of conditional sparsity as a core property of our approach. The conditional sparsity framework enables derivation of sharp weighted estimates and a mixed-norm estimate \( A_p^\alpha A_r^\beta \) that improves upon known sharp \( L^p \) bounds. Moreover, we develop dedicated sparse domination specifically for Doob's maximal operator, recovering the sharp bound as a direct application. Finally, we focus on the application of sparse theory to quantitative two-weight estimates.

An intuitive method for dimensionality reduction is proposed, which is highly effective for finding interesting projections of multivariate data. Following similar intuitive motivation to a number of existing techniques, the proposed method is based on enhancing the nearest neighbour relationships in the data. The proposed projection arises from the spectral decomposition of a matrix designed to encode the local covariance structure in the data, where the local covariance at a point is captured by pairs of its nearest neighbours. We show that under standard regularity conditions this matrix is a consistent estimator of the so-called “Density Information Matrix” (DIM); a non-parametric analogue of the Fisher Information Matrix. Spectral decompositions of DIMs have been shown to be connected with the important problems of Independent Components Analysis and, in the supervised context, Sufficient Dimension Reduction. However, existing estimators of the DIM are computationally expensive to compute and only target the DIM of a surrogate density, which is proportional to the square of the true underlying density. In addition, we go on to explore the practical utility of our method in aiding the downstream tasks of cluster analysis and outlier detection.

This paper proposes a nonparametric Bayesian inference framework for partially identified discrete response models. The key observation is that these models map a reduced-form conditional choice probability to an identified set. Consequently, nonparametric Bayesian inference for the conditional probability mass function leads to Bayesian inference for the identified set. The inference framework nests conditional moment inequalities and linear systems with unknown coefficients as special cases. Importantly, our proposal does not require converting conditional moments into unconditional moments or discretizing covariates. We show that the posterior is consistent for the true identified set when the model is correctly specified, show that the posterior can consistently detect model misspecification, and show posterior consistency for a pseudo-identified set that is valid under misspecification. We also verify the assumptions for a class of priors based on Gaussian processes that we use to implement our proposal. These priors offer similar flexibility to frequentist partial identification methods, and are computationally attractive because posterior sampling can be performed in closed-form. We also show that many of the ideas in this paper extend to continuous responses and aggregated discrete responses (e.g., market shares).

Monotone rearrangements provide a simple way to enforce shape constraints of an estimator, but existing distributional theory does not cover flat regions, where the target induces no local ordering. We study rearranged estimators in two canonical flat settings. First, for a histogram estimator of the uniform density, we establish functional weak convergence of its non-decreasing rearrangement at the parametric rate on compact subsets of the interval $(0,1)$ after an additional deterministic centering. This result is strikingly different from what is known for strictly monotone densities. Second, we consider two rearranged estimators of rearranged copulas under independence, based on empirical-copula increments and on a checkerboard approximation. After appropriate centering and rescaling, both estimators converge weakly on $[0,1]^2$ to an integrated Gaussian process which was not known before. We further use these results to prove asymptotic normality for a broad class of rearranged copula-based dependence measures, which were recently discussed in Strothmann et al. (2024).

Kolmogorov-Arnold Networks (KANs) replace fixed activations in deep architectures with learnable univariate edge functions, making the choice of edge parametrisation central. Existing variants rely on fixed bases such as splines, polynomials, or Fourier features, which impose a function-space geometry before data are observed. We introduce geometry-constrained KANs, a family of edge activations derived from Banach duality maps in which the geometry itself is learned through a scalar exponent $p > 1$ per edge. This exponent controls the qualitative response: sub-Euclidean values produce sharp, threshold-like behaviour reminiscent of the $\ell_1$ (LASSO) geometry, $p = 2$ recovers the linear regime, and larger values produce flatter responses near the origin. Across 50 symbolic-regression targets ($40$ from the AI Feynman benchmark plus $10$ synthetic stress tests), geometry-constrained KANs match or beat every fixed-basis baseline on median NRMSE (Banach-KAN $0.030$, tying Chebyshev and improving on splines); on average rank Banach-KAN is best on the $18$-equation core ($2.00$) and statistically tied with the strongest spline on the full benchmark ($2.32$ vs. $2.34$). The clearest gains appear under measurement noise: as $\sigma$ grows from $0$ to $1$, $\ell^p$-KAN degrades only $3.7\times$ – below even a cross-validated spline ($\approx 11\times$) – while an unregularised spline degrades $21.6\times$; Banach-KAN degrades $8.8\times$, comparable to a tuned spline but far more stable than the unregularised one. Banach-KAN also takes the most per-equation wins in the small-sample regime, with fixed-basis models catching up only as the training set grows. Learned exponents provide an interpretable, relative signal: at a fixed initialisation they reveal a consistent, target-dependent geometric ordering across equation families and input dimensions.

Epidemic models often pool air travel, commuting, and other mobility layers into a single weighted network. Pooling keeps the total imported infections into a region but discards the transport mode and route that delivered them, the information a mode-specific intervention needs. We make this precise for directed multilayer flows with heavy-tailed variation. Pooling acts as an asymmetric filter. The magnitude of an extreme importation, its radial tail index, is an exact invariant of sampling and nonnegative aggregation, whereas its composition across layers and routes, the angular part, is reweighted by layer-specific sampling and can be irrecoverably merged. We give a necessary and sufficient condition for recovering route composition, attach a surveillance decision cost to the loss, and show that the first established route need not be the busiest. A controlled metapopulation study calibrates the cost, and two contrasting reconstructions show where it bites. In US pandemic influenza, air adds fitted information beyond commuting, and a layer-resolved ranking targets more air-import burden than a pooled one. In the early Italian COVID-19 wave, commuting improves onset reconstruction while air attribution stays unresolved. Pooling can therefore support total-importation surveillance but cannot in general identify the mode or route behind an importation.

A frequency domain bootstrap procedure for functional time series is proposed and applied to the class of spectral mean operators. The procedure works by first generating independent pseudo periodogram operators across the positive Fourier frequencies using an estimator of the spectral density operator involved. Functional replicates of the spectral mean operators of interest are then generated. Through an additive, projection-based decomposition of the bootstrapped spectral mean operator, its leading $m$-dimensional part is properly complemented to also capture the relevant fourth order characteristics of the process. The complementation is achieved by means of a resampling procedure based on convolved periodogram operators of subsamples. The resulting bootstrap spectral mean operator consistently estimates the entire second order as well as the $m$-dimensional fourth order structure of the distribution of spectral mean operators. By allowing for the decomposition parameter $m$ to increase to infinity as the sample size increases to infinity, consistency in estimating the entire fourth order structure of the process also is achieved. The asymptotic theory developed investigates properties of the procedure for fixed and for increasing $m$ and establishes validity of the frequency domain bootstrap proposal under rather weak conditions on the underlying functional process class.

A key result in statistics is the data processing inequality, originally proved by Blackwell (1951) and later refined by DeGroot (1962) in terms of statistical uncertainty. The latter statement claims that the Bayes risk achieved on raw data always undercuts the Bayes risk on a processed version of the same data. This seemingly contradicts empirical findings in machine learning: pre-training, representation learning, feature learning, data augmentation are techniques used to improve performance of machine learning models. We reconcile both worlds by simply accepting that machine learning problems are constrained learning problems: the model class used does not include all measurable functions. We present counterexamples showing that the classical data processing inequality fails to hold in such a setting. Hence, we formulate a generalized data processing inequality, requiring the constrained Bayes risk of a joint distribution (with respect to a loss function and a constrained model class) to lower bound the constrained Bayes risk on the stochastically modified data distribution, regardless of the choice of distribution. We show this inequality to be equivalent to a set containment condition on a specific function set induced by the loss and model class, called the superprediction set. Finally, we exploit our characterization, derive sufficient conditions for this containment and quantify the inequality-gap.

Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the design variable is used as the error criterion. An initially randomly pruned fully connected deep neural network with logistic squasher as activation function is fitted to the data via gradient descent, using a data-dependent choice of non-constant stepsizes during gradient descent. It is shown that this network achieves (up to a logarithmic factor) the optimal minimax rate of convergence in case that the regression function is $(p,C)$–smooth. Here the estimate is able to circumvent the curse of dimensionality provided the predictors are concentrated in the neighborhood of a low dimensional manifold. The finite sample size performance of the estimate is illustrated by applying it to the simulated data.

Estimation of a regression function from exponentially $\beta$-mixing data is considered. The $L_2$ error with integration with respect to the design is used as the error criterion. Deep neural network estimates with logistic activation function are defined, where all parameters are learned by gradient descent. The rate of convergence of the expected $L_2$ error is analyzed for $(p,C)$-smooth regression functions. In the special case that the design is concentrated on a $d^*$-dimensional manifold, it is shown that the expected $L_2$ error of the estimate achieves a rate of convergence which depends on $d^*$ and not on the dimension $d$ of the design.

We address the simultaneous prediction of multiple high-dimensional physical fields governed by linear equality constraints, a setting that arises in many real-world applications in physics machine learning. Gaussian process (GP) regression is a widely used surrogate modeling approach due to its effectiveness in small-sample regimes and its ability to provide uncertainty quantification. However, applying GP models in this setting raises two major challenges: the high dimensionality of the discretized output fields and the enforcement of the physical constraint in predictions. For the latter, a common strategy consists in deducing one output from the others via the constraint relation. Through a benchmark, we show that this deductive approach is sensitive to the arbitrary choice of which output to deduce, affecting both predictive accuracy and uncertainty quantification. Consequently, there is a need for an approach that treats all fields symmetrically while strictly respecting the underlying physics. Motivated by these limitations, we propose a robust framework for jointly modeling constrained multi-field data. Our approach first leverages a specific PCA procedure for multi-field data, coined row-wise PCA, which has the interesting property of preserving the constraint in the latent space. Since standard PCA strategies for multi-field data do not preserve such constraints, we investigate theoretically the optimality of the row-wise choice. In a second step, we consider a linearly-constrained multi-output GP approach based on a specific kernel parametrization which is trained on the latent space of row-wise PCA. The proposed framework is validated on a population dynamics problem and on an industrial CFD application, which involves the prediction of Reynolds stress tensor components under the incompressibility constraint.

Exact Bayes prediction enjoys fast predictive regret guarantees, but exact posterior updating or representation may be too costly for online use. We study when these statistical guarantees are preserved by computational approximations. We show that the cumulative price of posterior approximation can be governed by the interaction between the contraction radius of the exact Gibbs posterior and the Wasserstein distance between the approximate and exact posteriors. Our general theorem shows that whenever exact Bayes prediction achieves a fast regret bound, any approximate posterior method that tracks the exact posterior with sufficient accuracy inherits the same fast regret, up to an additive term determined by the approximation error. Three online learning examples are developed. For linear models with strongly convex regularized losses, a projected Langevin algorithm yields an approximate posterior that achieves logarithmic regret. For an infinite-dimensional canonical exponential family sequence model over a Sobolev ellipsoid, a prior-preserving truncation method attains the minimax predictive regret rate with sublinear memory and constant update cost per observation. For random-design Gaussian process (GP) regression, a sparse variational posterior with inducing variables achieves the same predictive regret rate as the exact GP, but at substantially lower computational cost.

The simple exclusion process (SEP) is a paradigmatic model for nonequilibrium transport, yet its rich dynamics over an exponentially large configuration space remain notoriously intractable. Here, we leverage variational autoregressive networks to characterize the nonequilibrium dynamics of the symmetric (SSEP), asymmetric (ASEP), and totally asymmetric (TASEP) cases in one to three dimensions at any time. We validate the approach against established finite-time 1D and long-time 2D results for the SSEP, and further provide new results on dynamical activity, density fields and nonequilibrium phase transitions for 2D and 3D cases. In 2D, we give a directional-density criterion connecting the bulk-density organization to the 1D TASEP phase diagram, and reveal how boundary and bulk rates separately control the activity and susceptibility. In 3D, we provide the first finite-time characterization of the SSEP active-inactive phase transition, and uncover scaling relations over time. Across dimensions, the critical field follows a trend $s_c\sim L^{-2}$, suggesting tuning strategies by the characteristic diffusive length scale. Overall, this work establishes a unified neural-network framework for characterizing nonequilibrium dynamics of representative transport systems.

Spatiotemporal ARCH models capture temporal volatility persistence and cross-sectional dependence but typically require a predefined spatial weight matrix. This is restrictive in financial markets, where the dependence network is rarely known. We develop a LASSO-penalised quasi-maximum likelihood estimator that jointly learns a sparse weight matrix and estimates temporal dependence and covariate effects. Monte Carlo experiments show that the method recovers the model parameters and underlying network, with accuracy improving as the temporal sample size increases. We apply the method to daily returns from twenty UK-listed firms and compare the learned network with Euclidean-distance, correlation, autoregressive-similarity and sector-based structures. The learned network improves out-of-sample volatility prediction and reveals directional firm-level and cross-sector dependence not captured by the predefined alternatives. The method provides a data-driven framework for learning interpretable conditional-volatility networks.

Stochastic gradient descent (SGD) is typically analyzed at a deterministic horizon chosen before the algorithm is run, even though practical stopping decisions are made adaptively by inspecting the evolving trajectory. This mismatch creates a fundamental certification problem: fixed-time guarantees do not generally remain valid at data-dependent stopping times, while deterministic horizons derived from worst-case bounds can be highly conservative. We address this problem for strongly convex stochastic optimization by constructing fully observable, trajectory-adaptive upper confidence sequences for the squared distance of the last iterate to the optimizer and the suboptimality of a weighted average. These bounds hold simultaneously over time, attain the optimal $1/t$ decay rate up to iterated-logarithmic factors in the worst case, and adapt to the realized stochastic gradients, allowing SGD to stop as soon as a prescribed accuracy is certified without sacrificing statistical validity. Our approach treats the evolving SGD trajectory as a sequential experiment whose observations provide evidence about the unknown optimization error. To formalize this perspective, we develop new recursive confidence-sequence techniques and a general time-uniform empirical Bernstein inequality for adapted processes with time-varying conditional means and predictable ranges that may grow without bound. We further extend these confidence-sequence constructions to minibatch SGD, with the empirical Bernstein bounds exploiting the realized second-moment structure within each minibatch. Numerical experiments show that the resulting stopping rules can require several orders of magnitude fewer iterations than natural deterministic horizons.

Random feature methods provide a scalable approximation to kernel ridge regression (KRR), but the regularization parameter that yields the oracle learning rate depends on unknown smoothness and capacity parameters. In this work, we propose a neighboring early-stopping rule for adaptive regularization in KRR with random features (KRR-RF). The method uses a grid that is uniform in inverse regularization and compares only adjacent estimators, reducing the number of discrepancy comparisons relative to standard all-pairs Lepskii-type procedures. Both the neighboring discrepancy and its empirical complexity term can be computed directly in the random feature space, without constructing the exact kernel Gram matrix. We establish a high-probability comparison bound for neighboring KRR-RF estimators and show that, under standard source and capacity conditions together with suitable grid and random feature budget conditions, the selected estimator attains the oracle polynomial learning rate up to logarithmic factors. The result allows the regularization parameter to be selected without prior knowledge of the source and capacity exponents and covers both well-specified and partially misspecified regimes. Our analysis is based on an empirical random feature effective dimension that connects the observable stopping threshold with the population complexity of the random feature model. Simulation and real-data experiments illustrate the prediction performance and computational behavior of the proposed method in comparison with standard tuning procedures.

Functional data analysis is an important statistical field that treats data as random functions. In practice, the random functions are often not fully observed but instead measured at discrete times. While simpler problems, such as mean and covariance estimation, have been widely studied for discretely observed data, optimal estimation of linear regression for this data type has remained unsolved for over two decades. To tackle this fundamental challenge, we propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects. This unified estimation framework enables us to achieve minimax optimality in prediction risk in arbitrary sampling schemes ranging from sparse to dense designs, for both scalar-on-function and function-on-function regression models. Such methodological and theoretical advances are obtained for the first time and accurately reveal the influence of discrete sampling. For scalar-on-function regression, the phase transition occurs once, separating the convergence behavior into two distinct regimes. Remarkably, for function-on-function regression, up to three phase transitions may occur, determined by the sampling frequencies of the predictor/response functions. Finally, simulation experiments and two real data examples provide empirical support for the proposed methods.

We establish non-asymptotic Berry–Esseen bounds and bootstrap approximations for the Hilbert-space norm of nondegenerate $U$-statistics. Our triangular-array framework allows the kernel to take values in a separable Hilbert space $H_n$ that may vary with the sample size, thereby covering both infinite-dimensional spaces and Euclidean spaces of increasing dimension. The Gaussian approximation combines a Hilbert-space version of Stein's method with refined control of the higher-order terms in the Hoeffding decomposition. Its error depends on fourth moments of the kernel and on the spectral geometry of the covariance operator of the Hájek projection, particularly the proportion of squared spectral mass lying outside the leading eigendirection. Consequently, the theory accommodates singular and approximately low-rank covariance operators, provided that sufficient spectral mass remains beyond the leading direction. For $H_n=\mathbb{R}^{d_n}$, we obtain dimension-explicit bounds under coordinatewise fourth-moment conditions. We also establish approximation bounds for the empirical, Gaussian-weighted, and jackknife multiplier bootstraps, yielding tests with asymptotically correct size and consistency against alternatives at covariance-dependent separation rates. For testing the vector of pairwise Kendall's tau coefficients, a matching minimax lower bound shows that the resulting separation rate is minimax rate-optimal over dense Gaussian correlation alternatives.

Bayesian methods are essential in modern data analysis in ecology and evolutionary biology. They provide a flexible framework for modeling complex data-generating processes, while quantifying uncertainty based on the classical subjective interpretation of probability. However, Bayesian inference may provide misleading measures of uncertainty, particularly when the fitted model fails to adequately represent the true data-generating process. Although nonparametric approaches such as leave-k-out diagnostics and bootstrap resampling offer more robust alternatives under model misspecification, their computational cost is often too demanding because they require repeatedly refitting the same Bayesian model. In this paper, we introduce posterior covariance sensitivity analysis (PosCoSeA), a computationally efficient strategy for approximating leave-k-out diagnostics and bootstrap resampling without repeated model refitting. The performance of the methods is evaluated through both simulation studies and an application to real ecological data. We also provide an R package that implements these methods to facilitate their practical application.

We propose an omnibus nonparametric test for structural changes in the multivariate volatility matrix. The test aggregates bounded generalized quantile scores over a range of quantile levels and has a weighted leave-$q$-out $U$-statistic representation. Deleting nearby index pairs renders the centering effect induced by serial dependence asymptotically negligible. All quantities required for implementation, including the variance estimator used for standardization, are constructed under the null, without specifying volatility dynamics under the alternative. The standardized statistic converges to a standard normal distribution. We establish consistency against fixed alternatives that generate a positive integrated quantile-score signal and derive nontrivial local power against smooth departures and increasingly sharp transitions approaching multiple structural breaks. The bounded-score construction avoids the finite fourth- or eighth-moment conditions commonly imposed by least-squares and quasi-likelihood procedures, while aggregation across quantiles uses more distributional information than single-quantile methods. Monte Carlo results show satisfactory size and favorable power under heavy-tailed innovations, with competitive performance under Gaussian innovations. An application to the Fama–French three-factor model provides evidence against stability of the factor covariance matrix over the full sample and several economically relevant subsamples.

Multinomial choice models allow flexible substitution patterns but become computationally demanding with many alternatives or observations. With a fixed per-observation simulation budget, simulated maximum likelihood introduces simulation bias, while each optimization step requires a full-sample likelihood evaluation. We propose Stochastic Approximation with Unbiased Simulated Scores (SAUSS), an averaged stochastic approximation based on conditionally unbiased mini-batch score estimates. Each iteration uses a fixed mini-batch regardless of sample size. For multinomial probit, accept-reject sampling provides exact conditional draws and unbiased score estimates for any fixed number of accepted draws. Under local conditions, asymptotic theory for the averaged estimator and the partial-sum process of the SAUSS iterates incorporates mini-batch and simulation variability and supports random-scaling and plug-in inference. In simulations and an application, SAUSS gives comparable results in less than 1% of the computation time of simulated maximum likelihood. SAUSS extends to limited dependent variable models with conditional-expectation score representations and exact conditional sampling.

Comparing correlation matrices across time or stress scenarios is critical in quantitative finance and multivariate statistics, yet sample estimation noise often obscures whether an observed distance reflects a true structural shift. We derive the asymptotic sampling distribution of the intrinsic off-log (log-Euclidean) distance between two independently estimated full-rank correlation matrices under the null hypothesis that their population correlation matrices coincide. Under general sampling with finite fourth moments, the scaled squared distance converges to a weighted sum of independent $\chi_1^2$ variables, with weights determined by the asymptotic covariance of the Generalized Fisher Transformation (GFT) coordinates. Under Gaussian sampling at independence, this simplifies to a parameter-free $4\chi_d^2$ law. To calibrate tail probabilities, we provide closed-form cumulant generating functions, Lugannani–Rice saddlepoint quantiles, and an explicit Chernoff envelope requiring no root-finding. The first moment of the limiting law establishes a simple rule of thumb for the baseline expected distance under the null hypothesis ($\operatorname E[d_{\mathrm{LE}}] \lesssim 2\sqrt{d/n}$ near independence), quantifying the average separation induced strictly by estimation error. We establish plug-in consistency, present an explicit Gaussian covariance factorization, compare the distance statistic with coordinate Wald tests, and characterize its local power.

Paradigmatic interaction models explain how collective behaviors can emerge in complex systems from interactions among the constituent agents. In bio-inspired swarms, however, interactions alone may not suffice to bring the population to a desired aggregate configuration within a prescribed time horizon, as needed in applications ranging from targeted therapy to collective transport and emergency evacuation. In the present work, we consider finite-horizon minimum-energy collective steering for inertial swarms that are subject to stochastic disturbances. We focus on the mean-field representations of these multi-agent systems driven by Cucker–Smale alignment or Morse attraction–repulsion interactions. Our objective is to steer the swarm between prescribed endpoint distributions using a state-feedback control, where the endpoint specifications can be full phase-space distributions (positions and velocities) or position marginals alone. Our formalism is rooted in the theory of Schrödinger bridges, which has inspired contemporary developments spanning statistical inference, biological modeling, stochastic control, and generative learning. Within the bridges framework, the uncontrolled interacting stochastic dynamics are viewed as a prior model, and the optimal control as the minimum-energy corrective drift needed to realize the prescribed distributions. We derive nonlinear, coupled necessary optimality systems with a time-symmetric structure reminiscent of classical Schrödinger bridges, and propose nested fixed-point schemes to numerically solve them. Numerical examples show that the obtained optimal control (corrective drift) can dynamically exploit or counteract the interaction forces, depending on whether the latter are favorable or adversarial to the steering task.

Exponential random graph models (ERGMs) describe dependence among network ties, but inference becomes difficult when the likelihood is intractable and candidate network statistics are strongly correlated. We introduce BERGM Elastic Net, an adaptive empirical-Bayes approach that combines lasso shrinkage with ridge stabilization in a Bayesian ERGM. A latent-variable formulation supports approximate exchange sampling, while empirical-Bayes updates adapt the amount of regularization to the observed network. We connect the proposed prior to elastic-net penalized likelihood and clarify the interpretation of thresholded reporting and coefficient grouping. The method is developed for over-specified network models containing many related structural and covariate effects.

The OBABO and BAOAB schemes and the other standard Strang splittings of kinetic (underdamped) Langevin dynamics are widely used Markov chain Monte Carlo algorithms. Under a suitable friction scaling, the underlying diffusion relaxes on a ballistic time scale, suggesting that these discretizations, suitably tuned, sample targets with condition number $\kappa$ in $O(\sqrt{\kappa})$ iterations. We prove that no fixed choice of step size and friction, based only on the curvature bounds and the dimension, achieves this acceleration: total variation mixing time lower bounds for OBABO show that ballistic cold-start mixing fails uniformly over the smooth strongly convex class, and the lower bounds extend, with the same orders, to BAOAB and the other four Strang splittings. The proof transfers non-acceleration from optimization to sampling. Eliminating velocity gives an exact noisy heavy-ball recursion, and by the non-acceleration theorem of Goujaud, Taylor and Dieuleveut, for every tuning either some Gaussian target has a mode with relaxation time at least of order $\kappa$, or an attracting cycle exists on a smooth potential; dilating such a potential as $U_R(x)=R^2U(x/R)$ preserves its curvature bounds and produces metastability for a number of steps exponential in $R^2$, from an initial state at Wasserstein distance $O(R)$ from equilibrium. Using contraction estimates of Leimkuhler, Paulin and Whalley and a Wasserstein-to-total-variation regularization estimate, we prove a complementary upper bound of $O(\kappa)$ steps, up to logarithmic factors, for a fixed-parameter OBABO tuning. Hence, among fixed-parameter OBABO tunings, the optimal condition-number dependence of cold-start total variation mixing over this class is linear, up to logarithmic factors. A direct Gaussian calculation also rules out fixed-parameter acceleration for the left-endpoint exponential integrator.

We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by $n$ inequalities and a convex $L$-Lipschitz potential, this yields warm-start mixing in $\widetilde O((d^{2}+dL^{2}R^{2})\log(w/\delta))$ steps for the regularized Lee–Sidford walk. For a spectrahedron with $n\times n$ blocks, the log-det walk mixes in $\widetilde O((\psi^\star nd+dL^{2}R^{2})\log(w/\delta))$ steps, where $\psi^\star$ measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded $O(1/d)$-accurate metric computed from high-precision Lewis weights. For spectrahedra, given $\widehat\psi\ge\psi^\star$, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with $\psi^\star$ replaced by $\widehat\psi$ in the mixing bound.

Modern medical research, such as dose-finding studies, seamless trials, and shared control designs, often involves comparing multiple treatments simultaneously. Despite its wide applications, most research focuses on continuous endpoints, leaving the inference for general outcome types in high demand. In this article, we propose a new inference method for conducting multiple-treatment comparisons involving endpoints within the generalized linear model (GLM) framework under covariate-adaptive randomization (CAR). First, we investigate the asymptotic properties of the standard Wald z-statistics (z-scores) in multi-arm trials, highlighting issues when the working model is misspecified, particularly through omitted covariates. Our theoretical findings reveal that these \textcolor{black}{z-scores} do not consistently converge to a standard multivariate normal distribution, leading to either conservative or inflated Type I error rates, depending on the specific GLM endpoint. Second, based on these theoretical results, we develop adjusted test statistics to correct the distributional problems. To appropriately control the family-wise Type I error rate inherent in multi-arm comparisons, we incorporate our adjusted statistics with Simes-type multiple-testing procedures. This robust inference method can effectively control Type I error while potentially improving power. Extensive simulation studies and a real-world application to a metastatic breast cancer trial confirm the effectiveness and practicality of our approach.

We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $\kappa_\nu\in[1,d+1]$ denote the normalized trace-zero frame spread; $\kappa_\nu=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\le c_2d^2/\kappa_\nu$, the minimax mean-squared error is at least $c_1\log^2\{d^2/(\kappa_\nu n)\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$; hence fixed uniform accuracy requires $\Omega(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.

Direct numerical simulation of magnetic reconnection is limited by the scale separation of resistive magnetohydrodynamics. At large Lundquist numbers $S$ the current layer thins as $S^{-1/2}$, forcing fine grids and short time steps that make parameter scans prohibitively expensive. Deep learning neural operators offer an alternative by learning the map between function spaces rather than individual solutions, so that a single trained model returns the state for any parameter and time at inference cost. We have developed a Fourier Neural Operator (FNO) based Physics-Informed Neural Operator (PINO) surrogate for two-dimensional compressible, viscous, resistive reconnection in a wall bounded domain. We condition on the initial state, the Lundquist number, and a continuous query time. Predicting the magnetic flux function makes $\grad\!\cdot\!\bB=0$ exact, direct time queries remove autoregressive error accumulation, and parity-aware spectral differentiation matches the wall treatment of the reference solver. Trained across $10^{3}\le S\le2\times10^{5}$ with two values withheld, the surrogate recovers density, pressure, and guide field to better than $0.2\%$, velocity to $3.2$ to $3.9\%$, the flux function to $1.0\%$, and the spectral current density to $7.5\%$. It resolves the thin, narrow current sheet, Alfvénic jets, and spectral resolution beyond mode cutoff. Our surrogate reproduces the layer-averaged reconnection rate to within $3\%$ through $S=1.1\times10^{4}$, recovering a reconnection time exponent of $0.497$ consistent with Sweet-Parker scaling. When queried zero-shot on the $2049^{2}$ solver mesh, a full trajectory run is on the order of seconds on one GPU against several hours for the DNS, over two orders of magnitude faster, establishing learned operators as a practical route to surveying reconnection across parameter space.

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\log^2\{n/(4d)\}$ when $d\ge C$ and $n\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\log^2(n/d)\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.