Papers

A probability coordinate chart is a continuous strictly increasing bijection that transports observations to the open unit interval, where averaging is always well defined. For barycentric inference, the first coordinate moment is returned through the inverse chart. Higher initial coordinate moments similarly generate initial Kolmogorov moments by pullback, while centred coordinate moments remain on the probability scale. We develop inference for these quantities under fixed, intrinsic, and estimated charts. For a fixed benchmark chart, the coordinate mean is the primary inferential object. Confidence intervals are constructed on the probability scale and then transported through the inverse chart, preserving the geometry and allowing asymmetry on the observation scale. Hoeffding's inequality also provides finite-sample distribution-free intervals. The intrinsic case requires different treatment. The empirical cumulative distribution function is not an admissible chart, and its natural generalised plug-in construction collapses exactly to a middle order statistic. Feasible intrinsic inference therefore reduces to median inference. This self-induced functional is distinguished from the frozen oracle construction, which has a different asymptotic variance. For estimated location-scale charts, the asymptotic expansion contains a calibration correction for the first-order effect of learning the chart from the same sample. This yields an influence-function variance estimator and a bootstrap procedure that recalibrates the chart in every resample. Joint covariance theory is developed for vectors of initial and centred coordinate moments, with initial Kolmogorov moments obtained by inverse-chart pullback. The framework separates variation on the probability scale, amplification by the inverse chart, and uncertainty from chart calibration.

We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $\Omega(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin (1983), we prove an $\Omega(n^{-1.6342})$ non-anytime lower bound and an $\Omega(n^{-1.2408})$ anytime lower bound. These improve the recent $\Omega(n^{-1.932})$ non-anytime lower bound of Ma and Chen (2026) and the $\Omega(n^{-4/3})$ anytime lower bound of Tsai et al. (2026), respectively. Both results continue to hold when the stepsizes may be negative. Our anytime lower bound also shows that the $O(n^{-\log_2(1+\sqrt{2})})$ rate of non-anytime silver schedules (Altschuler and Parrilo, 2025; Grimmer et al., 2025) is unattainable in the anytime setting. This establishes a strict separation between the two settings.

We used nested models as a framework for characterising the consequences of a selectively measured predictor in clinical prediction models. In this framework, a model containing predictors available in the target population is referred to as the restricted model, while the extended model additionally includes a selectively measured predictor. We show that non-invariance in the restricted model between selected patients and the target population decomposes into components due to omission of the additional predictor, potential residual non-invariance, and their interaction. This framework is extended to a predictor measured via multiple routes of selection resulting in collider structures. We further show that imputation based on the conditional distribution of the additional predictor in the selected population transfers the restricted-model non-invariance to the imputed extended model as imputation bias. In a forthcoming version, this framework will be illustrated using real world and simulated data. The framework provides a formal basis for understanding how selective predictor measurement affects generalisability of clinical prediction models using routinely collected health data.

Data-consistent inversion (DCI) constructs probability measures whose push-forward distributions agree with observed data, while iterative data-consistent inversion (iDCI) extends this framework to generalized stochastic inverse problems by enforcing multiple push-forward constraints sequentially. Although iDCI avoids the direct approximation of high-dimensional joint densities, its relationship to the original joint DCI solution has remained unclear. In this work, we establish this relationship through copula theory. Using Sklar's theorem, we derive a factorization of the DCI update into separate marginal and dependence transformations and show that the discrepancy remaining after convergence of the iDCI algorithm is entirely characterized by the copulas associated with the observed and predicted joint distributions. This characterization motivates a copula-transformed iDCI solution, and we prove that an exact copula transformation recovers the original DCI solution. We further establish convergence results for approximate copula transformations under converging sequences of reference measures and progressively enriched feasible sets. Numerical examples demonstrate how the geometry induced by the quantity-of-interest map governs the importance of the copula transformation, illustrate an adaptive reference-measure refinement strategy for improving computational accuracy under a fixed sampling budget, and demonstrate the progressive refinement of generalized stochastic inverse problems through heterogeneous, asynchronously acquired experiments.

Generative models have been studied experimentally and theoretically as priors for inverse problems such as compressed sensing. Recent work by Gunn et al. studied the use of generative priors with tunable complexity, where a family of generative priors with varying complexity is maintained and a specific complexity can be selected at inversion time. They demonstrated that lower reconstruction errors can be experimentally attained for a variety of inverse problems by appropriately tuning the complexity of the generative prior. In the present paper, we establish theory for compressed sensing in the setting of a tunable family of linear generative priors naturally related through their singular value decompositions. We prove that in noiseless Gaussian compressed sensing, the full-dimensional linear prior attains the minimum expected reconstruction error over the entire family of linear priors. Thus, in this idealized linear noiseless setting, tuning to a lower-complexity prior does not improve the expected reconstruction error. This result is in contract to the behavior of denoising, where lower complexity priors attain lower reconstruction errors due to a standard bias-variance tradeoff. This result indicates that the experimental benefits of tunability in compressed sensing with neural network priors arises due to nonlinearities in the generative models.

Dependence can change between variable blocks without changing correlation; serial dependence complicates calibration. We propose DOMIC, a rank-based detector using density-operator mutual information (DOMI). Unit-norm random features make sample-level partial traces exact, so prefix sums compute segment statistics at per-split cost independent of segment length; a Gram form uses the exact kernel. Permutation tests are finite-sample exact under pair or block exchangeability for windows and schemes fixed in advance; validity after data-driven scheme selection is unproved. For i.i.d. pairs with feature dimension, frequency draw and bandwidth fixed, we prove a weighted chi-square limit at independence for the rank-based random-feature statistic. An additive entropy cost with Holevo gains drives segmentation. DOMI attains higher localised power than rank, copula, kernel and distance baselines in most non-Gaussian and correlation-free settings; the Gram form leads or ties in all 16 non-Gaussian settings where any statistic exceeds 0.10. On a three-break benchmark, Holevo partitioning returns exactly three breaks in 47.5% of replicates, against at most 19% for calibrated baselines. Block-calibrated primary scans reject in all three pre-specified pairs of a US financial panel, with strongest stage-two evidence for stocks and Treasury yields. One of 27 weather candidates passes the multiplicity-corrected re-test; candidate selection is unadjusted.

In the presence of interference, where the treatment assigned to one unit can affect the outcomes of others, many causal estimands depend on the treatment-assignment policy under which the experiment is conducted. This policy dependence creates a fundamental challenge for off-policy estimation, where the goal is to estimate causal quantities under a hypothetical intervention policy different from the one used to collect data. We study this problem of off-policy estimation of causal effects for heterogeneous Bernoulli policies. By representing exposure-weighted potential outcomes in the biased Fourier basis of the experimental design, we construct, for any prespecified Fourier subspace encoding the assumed interference structure, the unique minimum-$L^2$ weight that transports every function in that subspace. Global and local inverse-probability weights, linear-interference weights, and no-interference weights are special cases. The weight variance is a structured chi-square distance between the experiment and target policies. When the assumed interference structure is misspecified, the introduced bias couples the omitted outcome spectrum with the corresponding policy-shift coefficients, yielding a sharp robustness bound and a bias-variance trade-off. A Fourier-neighborhood-overlap condition gives consistency under structured interference, and we state a Doob-martingale central limit theorem for off-policy estimators. As the variance is not identified, we derive identifiable bounds and associated conservative estimators of the variance. Simulations illustrate these theoretical results for the design and analysis of experiments under network interference and design mismatch.

We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize risk stability through the critical learning rate, defined as the largest learning rate for stable training, and obtain $\eta_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $\eta_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-\rho)\}$, and $\eta_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^\beta(1-\rho)\}$, where $B$ is the batch size, $\rho$ is the momentum factor, and $\beta>1$ is the capacity exponent. Within this admissible region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient, through power-law decay, to a noise floor. We then minimize the final-step risk over the admissible learning rates and momentum factors under a fixed data budget, yielding a three-regime batch-size phase diagram that reveals how the role of momentum changes with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the best small-batch data-scaling exponent, thereby enabling greater parallelism without sacrificing data efficiency. In contrast, Nesterov achieves better data efficiency in the large-batch regime because its look-ahead mechanism suppresses noise accumulation. Numerical experiments validate the predicted stability boundaries, risk dynamics, and batch-size phase diagram.

We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.

We establish the consistency and asymptotic normality of a two-step estimator of conditional expectiles in the context of conditional scale models. We first estimate the conditional variance parameters by quasi-maximum likelihood and then compute the unconditional expectile of the innovations using the empirical distribution of the standardized residuals. We show how replacing true innovations with standardized residuals affects the asymptotic variances of both conditional and unconditional expectile estimators. Finally, our empirical analysis reveals that conditional expectiles assess tail risk in cryptomarkets in a more robust manner than traditional quantile-based risk measures, such as value at risk and expected shortfall.

A central challenge in distributional regression is to allow the shape of the conditional distribution of the response variable to vary flexibly with covariates while retaining directly interpretable effects on its mean and standard deviation. We extend the penalized transformation model (PTM) family into a conditional-shape PTM, which assigns separate structured additive predictors to the conditional mean, standard deviation, and standardized distributional shape beyond location and scale. A covariate-dependent monotone transformation maps the standardized response to a fixed reference distribution, while affine standardization enforces mean zero and variance one for the induced standardized distribution. Thus, the first two predictors remain exactly the conditional mean and standard deviation. The shape predictor accommodates selected linear, nonlinear, group-specific, spatial, and interaction effects; regularization shrinks unsupported departures toward a reference-family location-scale model. We fit the PTM using mini-batched stochastic variational inference with model-aligned Gaussian blocks and staged optimization. In simulations, the PTM recovers smooth mean and standard-deviation effects and a covariate-dependent transition from skewness to bimodality while suppressing unnecessary shape effects. Under a deliberately misspecified design, it remains competitive with a structured additive Dirichlet-process mixture in test-set density and distribution-function accuracy, although all models show undercoverage and both flexible methods miss fine features. Applications to $13{,}425$ Norwegian water-conductivity observations and $1{,}182{,}514$ German daily-temperature observations demonstrate selective group-specific, seasonal, and spatial shape variation. Predictive performance criteria favor the conditional-shape PTM over a fixed-shape PTM and a Gaussian location-scale model.

Clinical trials often show treatment curves that diverge early and converge later, or vice versa patterns that are poorly captured by the proportional-hazards assumption. We develop a joint inferential framework for two nonparametric functionals of censored survival data: the Kaplan–Meier-based Mann–Whitney effect and a novel temporal contrast separating early and late differences. The approach provides interpretable, probability-scale effect measures and enables joint inference for global and temporal contrasts under right censoring. In simulation studies, the method outperforms the log-rank test under non-proportional hazards while maintaining nominal type-I error. A real-world application illustrates how the temporal contrast reveals clinically meaningful early treatment advantages that remain hidden in standard analyses

This paper investigates probability-maximizing sequential change detection, a formulation in which performance is measured by the probability of stopping within an admissible interval of duration $\xi$ after a change rather than by the expected detection delay. Earlier work introduced this viewpoint in a Bayesian setting and subsequently formalized it under Lorden- and Pollak-type minimax criteria for the case in which successful detection must occur on the first post-change observation, i.e., $\xi=1$. This paper generalizes this framework in two directions. First, the decision maker is allowed to stop within a prescribed window of $\xi\in\mathbb{N}$ post-change observations. Second, the monitored process is allowed to experience multiple, non-overlapping transient change episodes with unknown onset times and durations, so that success consists of stopping within the admissible window associated with any one of these episodes. False alarms are controlled through an average run-length constraint. For exact finite-sample analysis, the paper introduces a survival-weighted average success criterion, which represents the probability of successfully detecting a randomly encountered change opportunity conditional on the detector being active at its onset. It is established that this criterion admits an exact representation as the expected truncated Shiryaev–Roberts (TSR) statistic at the stopping time normalized by the average run length, and it characterizes its exactly optimal stopping rule. The optimal procedure has finite memory and aggregates the likelihood-ratio evidence corresponding to all possible change onsets within the most recent window and compares the resulting TSR statistic with a state-dependent continuation boundary obtained from an optimal-stopping formulation.

Process-based forest models are widely used to simulate forest growth and responses to environmental change, but their calibration and application often require many computationally expensive model evaluations. We present an implementation of the Physiological Processes Predicting Growth (3-PG) model in JAX that uses just-in-time compilation, vectorization, and GPU acceleration to reduce execution time. The implementation also supports automatic differentiation, providing gradients of model outputs and calibration objectives with respect to model parameters. This enables efficient gradient-based optimization and gradient-informed Bayesian calibration, extending 3-PG beyond conventional gradient-free approaches. The implementation produced results numerically consistent with r3PG for the evaluated configuration. Overall, the JAX implementation provides a faster and differentiable framework for calibrating and applying the 3-PG model.

The novel Frankenstein was published anonymously in 1818, and was first credited to Mary Shelley in a French translation of 1821. Since its publication, several claims - both contemporaneous and recent - have been made suggesting that Frankenstein was actually written by Mary's husband, Percy Bysshe Shelley. We review the background of this controversy and then apply modern techniques from computational stylometry to determine who the true author is. Based on our analysis, we find extremely substantial evidence that Mary Shelley is indeed the true author of Frankenstein, and that it is very improbable that Percy Bysshe Shelley played a heavy role in composing the text. While our finding confirms mainstream scholarly opinion regarding Frankenstein, our analysis is the first application of stylometric techniques to this question and provides strong objective grounds for favouring Shelley by freeing the question from some of the politics which have traditionally accompanied it.

Differential privacy (DP) has emerged in the computer science literature as a measure of the impact on an individual's privacy resulting from the publication of a statistical output such as a frequency table. This paper provides an introduction to DP for official statisticians and discuss its relevance, benefits, and challenges from a National Statistical Organisation (NSO) perspective. We motivate our study by examining how privacy is evolving in the era of big data and how this might prompt a shift from traditional statistical disclosure techniques used in official statistics–which are generally applied on a cell-by-cell or table-by-table basis–to formal privacy methods, like DP, which are applied from a perspective encompassing the totality of the outputs generated from a given dataset. We identify an important interplay between DP's holistic privacy risk measure and the difficulty for NSOs in implementing DP, showing that DP's major advantage is also DP's major challenge. This paper provides new work addressing two key DP research areas for NSOs: DP's application to survey data and its incorporation within the Five Safes framework.

Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.

In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.

A fundamental quantity in machine learning is the optimal performance achievable by any model on a given task. Estimating this quantity allows us to distinguish the irreducible part of the error from a deficiency of the model, telling us how much room for improvement remains. Recent work has shown that the Bayes error, or equivalently the optimal accuracy, can be estimated from soft labels in binary classification. However, accuracy is often a poor summary of performance in settings with severe class imbalance or noisy annotations, where metrics such as the balanced error rate (BER) and the area under the ROC curve (AUC) are more appropriate. We address this gap with two complementary contributions. (i) Estimation. We propose soft-label-based estimators for the optimal BER and AUC. We first consider the clean setting in which true soft labels and the class prior are known, and then extend the estimators to a more realistic setting in which the class prior is unknown and the observed soft labels are corrupted by an unknown order-preserving transformation, possibly followed by additive noise. In the latter setting, we approximately recover the clean soft labels via isotonic regression with auxiliary hard labels, estimate the class prior with a clipped mean of the hard labels, and derive finite-sample error bounds for the resulting plug-in estimators. (ii) Evaluation. Since the optimum is unobservable on real datasets, evaluating any such estimator is itself nontrivial. We extend the FeeBee framework, originally proposed for evaluating Bayes-error estimators, to the optimal BER and AUC. The resulting procedure provides practical evaluation scores without requiring knowledge of the optimum, and applies to any estimator of the optimal BER or AUC, not only our proposed ones. Experiments on synthetic and real-world datasets validate both the estimators and the evaluation procedure.

Learning graph structures from data is a fundamental problem that spans a wide range of signal processing and machine learning tasks. While significant effort has been made to tackle the problem, existing research has largely evolved along two parallel directions. The first seeks to infer the topology of an individual graph from observations supported on it, whereas the second seeks to learn a generative distribution from observed graph instances, enabling the sampling of new graphs. This review presents a unified framework that connects these formulations by viewing them as inverse problems of a common generation process for graph data. We review the major methodologies within this framework, highlight their relationships, strengths, and limitations, and identify opportunities for integrating ideas across paradigms. By bridging graph topology learning and graph generation, this review provides a broader cross-disciplinary perspective on the field and outlines promising directions for future research.

Homogeneous random graphs, also known as Erd{\H os}-Rényi graphs, are a subset of the family of dense random graphs, specified by a graphon. We analyze several goodness-of-fit tests for these models that are based on subgraph counts. To obtain the limiting null distribution of the test statistics we use a decomposition of graph functionals, which reveals a cancellation effect. Motivated by a quasirandomness result we obtain a test that is consistent against all alternatives, and we use two popular parametric subfamilies to evaluate the other tests. The theoretical results refer to the limit $n\to \infty$ of the size $n$ of the graph, the behavior for finite $n$ is illustrated by simulations.

Generative modeling directly on geometric manifolds can avoid errors introduced by flattening non-Euclidean data, repeated ambient projection, and coordinate inconsistency in Euclidean representations. Schrodinger bridges provide a probabilistic generative framework for entropy-regularized transport between prescribed endpoint distributions. We study Schrodinger bridges for kinetic dynamics on Lie group manifolds with state X_t = (g_t, xi_t) in G x g, allowing endpoint observations to constrain only the variables that are actually measured. In particular, the entropy projection determines the conditional law of the unobserved endpoint velocities. For the same observed endpoint bridge, we develop two computational realizations: Wrapped-Kernel Bridge Calibration (WKBC) uses an explicit periodized kinetic kernel on compact Abelian groups, whereas Reciprocal Conditional-Control Bridge Matching (RCCBM) handles compact non-Abelian groups through two-sided endpoint calibration and mollified conditional-control matching. The canonical teacher-mixture path law is itself a Markov reciprocal law, so forward generation uses a calibrated initial law and one learned Doob controller. Moreover, we establish a modular error bound in the bounded-Lipschitz path metric that provides a clean separation of errors due to endpoints, control regression, initialization, discretization, and related approximations. Experiments on multiple Lie group manifold datasets validate the feasibility and consistency of our proposed method, covering protein and RNA torsions, SO(3), U(n), and the Protein Conformational Transition Pathway Generation task using mdCATH trajectories in a compact reduced representation. The source code is publicly available at https://github.com/cafferyzhang12/Schr-dinger_Bridge_on_LieGroup.

The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($\Sigma=\sigma^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\to\infty$, its $k$th singular value satisfies $\ell_k\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\to\infty$ and $m/d\to0$, the expected Kendall correlation is $\frac{2}{\pi}\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\log n\ll m\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.

We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $\delta\in[0,1/4]$, batching gives $O(T^\delta)$ calls per round and $O(T^{4/5-\delta/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) methods enable scalable Bayesian inference, but their performance depends strongly on hyperparameters such as the step size, mini-batch size, and number of leapfrog steps. Since most SGMCMC algorithms lack a Metropolis-Hastings acceptance rate, standard acceptance-based tuning methods are not directly applicable. We propose HyperMC, a multi-fidelity tuning framework that combines Hyperband-style resource allocation with kernel Stein discrepancy (KSD) evaluation. By running multiple successive-halving brackets, HyperMC balances broad exploration of a continuous hyperparameter space with increasingly accurate evaluation of promising configurations under a fixed computational budget. We further introduce Robust HyperMC, which uses global grid initialization followed by elite-guided local refinement to reduce sensitivity to random candidate generation and noisy finite-budget evaluations. Under suitable approximation and concentration conditions for the estimated KSD, we establish that the successive-halving component selects a near-optimal configuration among the sampled candidates with high probability and derive a sufficient computational budget for successful selection. Experiments on logistic regression, probabilistic matrix factorization, and Bayesian neural networks show that HyperMC improves posterior approximation or predictive calibration relative to MAMBA, grid search, and heuristic baselines, while Robust HyperMC yields more stable and reproducible tuning results.

This paper develops a unified framework for instance optimal sparse recovery from nonlinear observations. The main ingredient is a signal-dependent restricted approximate invertibility condition (RAIC) of some gradient, which leads to the instance optimality of iterative hard thresholding. Under Gaussian designs, we apply the proposed framework to phaseless, one-bit, and ReLU measurements, which correspond to the problems of sparse phase retrieval, one-bit compressed sensing, and sparse ReLU regression, respectively. For sparse phase retrieval, we propose a variant of thresholded amplitude flow and show its instance optimality under $O(s^3)$ measurements (up to logarithmic factors), where $s$ is the sparsity level. To our best knowledge, this is the first instance optimal efficient algorithm for sparse phase retrieval and complements Gao, Wang and Xu (2016) that achieved this via a computationally intractable program. In one-bit compressed sensing, we establish the instance optimality of normalized binary iterative hard thresholding and strengthen the recent result of Matsumoto and Mazumdar (2024). In sparse ReLU regression, it is shown that a slight variant of the algorithm in Soltanolkotabi (2017) is instance optimal. Moreover, $(\ell_2,\ell_2)$ non-uniform instance optimal guarantees are obtained for these problems. The analysis is built upon a number of high-dimensional concentration bounds, including bounds on restricted eigenvalues and a novel instance-dependent hyperplane tessellation result.

Computationally expensive black-box models often involve a large number of input factors with complex interactions and varying importance. Experimental design techniques can be used for quickly identifying the important factors, which can make the optimization of a complex computer model or the training of an expensive machine learning model more efficient. Existing screening designs for black-box models focus mainly on continuous factors, with the maximum one-factor-at-a-time (MOFAT) design being a recent example. In this work, we extend the design to incorporate multiple types of factors, including nominal, ordinal, and discrete-numeric. We first identify the properties leading to optimal screening, where qualitative and quantitative factors should be treated differently. Based on these properties, we propose practical algorithms to efficiently construct MOFAT designs for all types of factors. The usefulness of the design is demonstrated by both numerical experiments and an application to hyperparameter tuning in machine learning models.

Factorial designs allow simultaneous estimation of multiple main effects and interactions, but covariate imbalance can substantially reduce estimation precision. Existing rerandomization methods improve covariate balance yet do not fully exploit heterogeneous priorities across factorial effects or effect-specific covariate importance. To address these limitations, this paper proposes a data-adaptive rerandomization framework for $2^K$ factorial designs. We first develop an oracle criterion that jointly incorporates researchers' priorities over factorial effects and effect-specific covariate importance, enabling precision gains with guaranteed lower bounds. To make the oracle criterion implementable, we develop a data-adaptive procedure that learns effect-specific covariate importance from a random subset of units and applies an estimated oracle criterion to the remaining units. Unlike existing two-stage rerandomization methods for treatment-control experiments, our procedure accommodates multiple factorial effects and requires no auxiliary dataset. Under a finite-population framework, we establish design-based asymptotic theory and show that the proposed procedure preserves the oracle design's precision-prioritization property and, under suitable conditions, achieves the same asymptotic precision as the oracle design. Numerical studies demonstrate substantial efficiency gains over existing rerandomization methods.

Experiments on networks observed over time face network spillovers, temporal carryover, and dependence deliberately introduced by the design. We propose COSTA—Covariance-Optimized Spatiotemporal Treatment Allocation—a joint Bernoulli design for unit–time assignments. Under common treatment marginals and a nonnegative linear network–temporal exposure model, Horvitz–Thompson bias for the sustained all-treated versus all-control contrast is exactly the negative expected weight of an assignment cut. A covariance-level variance envelope yields an MSE bound that can be optimized directly over assignment covariance. To scale this design, we introduce a thresholded-Gaussian Kronecker parameterization that mirrors the network and temporal exposure operators while preserving valid Bernoulli marginals. We next develop inference theory for the joint effects of designed treatment dependence and interference-induced outcome dependence. Canonical correlations between latent blocks generating separated HT contributions supply the coefficients required by graph-$\psi$ central limit and network-HAC theory; a spectral-floor and far-row-mass condition gives a primitive sufficient check. The framework covers sparse, block, Kronecker, locally factored, and other structured covariance sequences satisfying these conditions. Semi-synthetic RetailRocket and MovieLens experiments show substantial default-setting RMSE reductions and well-calibrated model-assisted design-centered intervals across linear, nonlinear, and demand-substitution outcome surfaces.

Motivated by the statistical inference using the Gram matrix in the context of missing at random observations, this paper investigates the spectral properties of the random matrices $\mb S_n=\frac{1}{n}\mb Z\mb Z^*$, where $\mb Z=\mb D\circ(\boldsymbol{\Sigma^{1/2}}\mb X)$ represents a Hadamard random matrix with entries determined by independent Bernoulli variables $\mb D$. Operating within the high-dimensional framework, we establish the convergence of the empirical spectral distribution of $\mb S_n$ to a well-defined limiting distribution. In addition, we explore the impact of the missing mechanism on the second-order properties of the spectral distribution of the Gram matrix $\mathbf{S}_n$. We establish the central limit theorem for the linear spectral statistics of $\mathbf{S}_n$, shedding light on their fluctuations. Surprisingly, our analysis reveals that even in the ideal Gaussian distribution scenario, the fluctuations of statistics generated by eigenvalues are influenced by the eigenvectors of the population covariance matrix in the missing-at-random case. This discovery uncovers a remarkable phenomenon that starkly contrasts with the classical case. Subsequently, we demonstrate the practical application of our central limit theorem in hypothesis testing for the population covariance matrix.