Papers

Exact Euclidean \(K\)-means partitions \(n\) observations into \(K\) unlabelled clusters, but the unrestricted search is generally exponential. We use data-derived geometric graphs to precondition an exact subset dynamic program: as a result only connected vertex subsets are admitted as clusters, while sum-of-squared-errors (SSE) loss is unchanged. A remaining-set recurrence minimises fixed-\(K\) or penalised SSE, with exact factorisation over the connected components of each remaining set. The central question we study is how much computational support can be removed while preserving an unrestricted optimum. Graph inclusion gives monotone coverage and support relations, and a bottleneck threshold identifies the first covering graph in a nested hierarchy. For fixed \(K\) and dimension, under compact ball support and density bounds, retaining \(q=O(\log n)\) nearest neighbours per observation preserves an empirical SSE optimum with probability tending to one, using an \(O(\log n/n)\) fraction of complete-graph edges. Truncated Gaussian mixtures with unequal weights and covariances satisfy these conditions. The rate we provide is a sufficient upper bound rather than a result implying polynomial optimisation complexity. Objective-matched synthetic and full-data comparisons assess coverage, compression, and reference-label agreement. As a secondary application, we illustrate how the proposed scaffold preconditioning can be utilized to improve the efficiency of split-merge proposals that preserve unrestricted mixture posteriors.

We investigate the least squares linear regression problem with random partial Discrete Fourier Transform (DFT) matrices, providing a rigorous analysis of the model's generalization error. By leveraging tools from random matrix theory, we derive exact non-asymptotic bounds for the risk of the Moore-Penrose estimator, which hold for finite-dimensional problems and reveal the precise dependence on key parameters such as the sample size, dimension, and noise variance. Then we obtain a characterization of the double descent phenomenon in the linear regression context, demonstrating how the risk evolves when the number of parameters $p$ and the number of samples $n$ tend to infinity, with $p/n$ fixed. The analysis relies on applications of the Stieltjes transform for random Fourier matrices, enabling a precise description of the spectral properties of these matrices and their impact on regression performance. To validate our theoretical findings, we present several numerical examples that illustrate the double descent curves. These simulations align closely with our derived bounds, confirming their predictive power in both under-parameterized and over-parameterized regimes.

This paper investigates the numerical resolution of the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures. We consider its application to the estimation of Gaussian mixture models (GMMs) with an unknown number of components and unknown diagonal covariance matrices. Our approach combines the Conic Particle Gradient Descent (CPGD) principle with Riemannian gradient descent, to account for the underlying Fisher-Rao geometry of Gaussian distributions. Our contributions are twofold. First, we provide theoretical guarantees for the convergence of our algorithm. In particular, we establish exponential local convergence under a non-degeneracy condition on the solution and relate this assumption to a separation condition on the underlying statistical target. Second, we address practical implementation aspects of CPGD and present numerical experiments illustrating its performance. On the test cases considered, these experiments suggest that CPGD is more robust to overspecification of the number of components than the EM algorithm. We also investigate the impact of component separation on recovery accuracy.

Statistical model transformations based on randomly stopped sums, maxima and minima are widely used to extend statistical models. We characterize the complete set of stopping models for which randomly stopped sum and extreme model transformations function as statistically stable (idempotent) model extensions. Stability requires the underlying stopping model to be closed under pgf composition. We prove that any finite-dimensional, connected stopping model closed under pgf composition is necessarily a family of random variables whose pgfs commute. Using the corresponding Koenigs function, we establish that these models form a statistical manifold admitting a global, one-dimensional parametrization $\theta = \Pr(N=1) \in (0, \theta_*]$, where the probability mass at $i$ is a polynomial in $\theta$ of degree at most $i$. Finally, we establish a duality between stopping models closed and containing the identity variable (the ones yielding stable extensions) and the set of probability distributions supported on the positive integers. These findings disprove the long standing conjecture that statistical stability occurs only under geometric stopping.

Thin plate splines are highly attractive smoothers. However, they have cubic computational cost, which severely limits their use in practice. As a remedy, Wood (2003) suggested thin plate regression splines (TPRS), which provide a low rank approximation. The key step of the TPRS approximation is a truncated eigendecomposition of the radial basis function (RBF) design matrix. However, as Wood (2003) writes, the optimality of the TPRS approximation is a slightly weak one. This is because the RBF coefficients are subject to orthogonality constraints and the TPRS approximation is only optimal if these constraints are ignored. To address this shortcoming, we suggest a slightly different low rank approximation. The suggested approximation is based on a truncated Demmler-Reinsch basis (TDRB), which provides a best low rank approximation of the smoother matrix in terms of Frobenius and spectral norm. We prove that the TDRB smoother achieves the optimal rate of convergence and suggest an efficient algorithm for its construction. This algorithm is based on a truncated Karhunen-Loève (KL) expansion of the equivalent Bayesian smoothness prior and it has the same computational cost as required for TPRS. We demonstrate the applicabilty of our approach through simulations and a real data example. We find that the performance is very similar to that of TPRS but the suggested approach has some advantages.

Under Gaussian sampling, removing location and scale turns the sample into a direction that is uniformly distributed on a sphere of dimension $n-2$, and sample skewness and kurtosis become a cubic-quartic polynomial image of spherical measure. We use this geometry to obtain exact finite-sample results for the Jarque-Bera statistic $\mathrm{JB}_n$. First, for every $n\ge 3$ the largest attainable value of $\mathrm{JB}_n$ is $n\{(n-2)^4+4(n-1)^2\}/\{24(n-1)^2\}$, attained by a single-outlier sample, so the asymptotic $10\%$ and $5\%$ tests have size zero for $n\le 6$ and the $1\%$ test for $n\le 7$; near this maximum the density of $\mathrm{JB}_n$ behaves like an explicit multiple of $(J_n^+-x)^{(n-4)/2}$. Second, passing from residual coordinates to power sums writes the joint density of skewness and kurtosis as an integral of the reciprocal square root of a polynomial discriminant; for every $n\ge 5$ the innermost integral runs over a single interval and is a Lauricella $F_D$ period. Third, this yields explicit laws: an arcsine law for $n=3$, an algebraic joint density for $n=4$, and a single complete elliptic integral, equivalently ${}_2F_1(\tfrac12,\tfrac12;1;\cdot)$, for $n=5$. Residual-coordinate collisions generate the discriminant singularities of the joint law, while stationary points of $\mathrm{JB}_n$ on the residual sphere govern the singularities of its one-dimensional density.

A quantum measurement transforms a quantum statistical experiment into a classical one, but different measurements generally yield experiments with different statistical information. We introduce quantum-to-classical sufficiency: relative to a prescribed measurement class, a measurement is sufficient if its induced classical experiment Blackwell-dominates those from every other measurement in the class. This separates two components of quantum inference: reducing the measurement class through symmetry or decision-theoretic arguments, and identifying a canonical measurement in the reduced class by an experiment-level sufficiency condition. We establish a factorization criterion where the parameter enters the group-averaged experiment only through weights of an orthogonal block decomposition. The block-label measurement is then sufficient relative to all invariant measurements. For tensor-product models under unitary conjugation, this identifies weak Schur sampling as the sufficient measurement for spectral inference. Combined with risk-preserving symmetrization, it reduces invariant Bayes and minimax problems over all measurements exactly to classical decision problems based on Young diagrams. We derive sharp first-order asymptotic Bayes and minimax risks for smooth spectral functionals. We establish an analogous reduction for a noncompact group action in a continuous-variable setting. We consider displaced thermal states and functionals of the thermal parameter with an unknown displacement. Within displacement-invariant measurements, total residual photon number measurement emerges as the sufficient measurement. Combined with Hunt-Stein reduction, this yields exact finite-sample reductions of Bayes-minimax and minimax problems over all measurements to classical decision problems based on the total residual photon count, which follows a negative binomial distribution.

A regular observed statistical model may converge to a limit in which a previously identifiable signed parameter becomes identifiable only modulo a reflection. We study the local information geometry of this transition. For a twice differentiable Hellinger embedding with an exact limiting reflection, the observed displacement is forced into the two-jet form \(\varepsilon\lambda J_-+\lambda^2J_+/2\), up to higher-order terms. The mixed jet restores the sign away from the symmetric face, whereas the even jet is the first tangent inherited by the quotient. After nuisance elimination, a positive Gram determinant yields a nondegenerate cross-cap two-jet. The associated local asymptotic theory has three regimes governed by \(\tau_n=\sqrt n\,\varepsilon_n^2\): regular signed LAN, a critical curved Gaussian subexperiment, and a quotient regime with the \(n^{-1/4}\) signed scale. We prove that the same parabolic critical experiment persists for predictive likelihoods along a single stationary dependent trajectory. The limiting quotient has a regular Fisher metric in the invariant coordinate, while its pullback degenerates in the signed coordinate. For a solvable CIR–OU benchmark motivated by coherent sea-clutter observations, we derive the quotient Fisher metric and curvature explicitly and show that the curvature is strictly negative. We also determine the restricted holonomy of the full Amari family: \(\operatorname{Hol}_0(\nabla^{(a)})=SO(2)\) for \(a=0\), whereas \(\operatorname{Hol}_0(\nabla^{(a)})=GL^+(2,\mathbb R)\) for \(a\neq0\). The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.

In this work, we study the one-dimensional regression problem under random design and Gaussian errors. Our framework is very general: we make no prior assumptions about the design (which may be nonstationary and exhibit short or long-range dependence), nor do we assume that the errors are homoscedastic. We examine in detail the cases where the error process exhibits short or long-range dependence. We adopt a least-squares penalized strategy using piecewise polynomials to estimate the regression function, following the framework of Baron, Birgé and Massart [1999]. We derive explicit penalties, up to calibration constants, to obtain adaptive estimators for which we establish risk bounds. Since these penalties depend on the dependence properties of the error process, which are unknown in practice, we propose several adaptations of the dimension jump calibration algorithm to make our procedures fully data-driven.

We establish moment and mixed-tail inequalities for fixed-order decoupled homogeneous chaoses generated by independent, centered, sparse bounded random variables. Our bounds apply to arbitrary real rectangular coefficient tensors and describe the fluctuation scales through weighted slice and partition norms, with a Bennett-type logarithmic improvement in the largest-entry term. As an application, we derive guarantees for sparse Khatri–Rao embeddings that explicitly account for sparsity and input geometry.

Adaptive arm selection changes the distribution of the observations collected by a bandit algorithm, but it need not change their limiting empirical spectrum. We study Gaussian bandit designs in which the dimension and the number of observations grow proportionally. A quantitative coupling theorem compares the design generated by any causal selection rule with an independent Gaussian design. If the logarithm of the number of available arms is sublinear in the dimension, the empirical spectral distribution converges to the Marchenko-Pastur law, uniformly over the selection rule. Consequently, Gaussian Bayesian bandits have policy-independent first-order limits for posterior mean-square uncertainty, squared posterior covariance, and information acquisition. For linear-score selection, we obtain the exact conditional arm distribution and show that two-arm selection produces an exactly Wishart Gram matrix in every dimension, despite its nonzero conditional mean. For a fixed selection direction, we identify an explicit eigenvalue and eigenvector transition governed by the second moment of a Gaussian maximum. A counterexample shows that a direction's overlap with a reference signal does not determine this transition. Finally, an exponentially large arm pool permits a different bulk limit, establishing the order-sharpness of the arm-growth condition. These results distinguish global spectral stability from directional effects in adaptive bandit data.

Matrix time series is a series of matrix data observed over time. Analytical tools for such time series is needed in many applications in finance, economics, engineering and many other fields. To avoid the use of vectorization of the matrices which loses the column and row information, and the vector autoregression framework in traditional time series analysis, \cite{chen2021autoregressive} proposed the Matrix Autoregressive (MAR) Model. The model maintains and utilizes the matrix structure, leading to a substantial dimensional reduction and admitting explicit interpretations, comparing with the vector autoregressive model on the vectorized data. However, the MAR model still encounters difficulties in dealing with large dimensional matrix time series as the coefficient matrices in MAR models are also large. In this paper we propose to achieve further dimension reduction through reduced-rank constraints of the coefficient matrices in the MAR model. Estimation and rank determination procedures are studied. Theoretical investigation and empirical examples show that the reduced-rank constraint can achieve higher statistical efficiency than the MAR model.

We initiate the study of approximating the top eigenvalue and eigenvector of a random symmetric matrix $ A \in \mathbb{R}^{n\times n} $ using $ q(A)b $ where $q$ is a degree-$d$ polynomial and $b$ is a standard Gaussian vector independent of $A$. For spiked GOE $ Y = \lambda vv^\top + X $, we identify $ d_\star = \frac{\log(n)}{2\log(\lambda)} $ to be the critical degree threshold above which accurate approximation of the top eigenvalue and eigenvector is possible. This sharpens the common belief that spectral methods can be implemented by $ O(\log(n)) $-step power iterations and offers a precise connection between spectral methods and low-degree polynomial algorithms, a popular proxy for all polynomial-time algorithms. For GOE $X$, we identify $ d_\star = n^{1/3+o(1)} $ to be the critical degree threshold for top eigenvector approximation, whereas constant degree suffices for top eigenvalue approximation. Moreover, in the limit where $ d/n^{1/3} $ converges to a positive finite constant, we compute the exact asymptotic eigenvector approximation accuracy in terms of the expected squared overlap. These results significantly improve upon predictions made in randomized numerical linear algebra for deterministic data matrices that the iteration count of power methods with random initialization is governed by the inverse spectral gap. Technically, our analyses leverage extremal properties of Chebyshev polynomials and draw upon the rich literature of random matrix theory.

This note presents a general form for a proper scoring rule that can be used for possibly right-censored survival data without requiring any knowledge or assumptions about the censoring process.

We study Bayesian linear bandits with an isotropic Gaussian parameter, independent Gaussian candidate arms, and Gaussian reward noise when the horizon is proportional to the dimension. The normalized posterior uncertainty has an explicit limit that is uniform over all causal policies. Gaussian posterior identities then determine the limiting parameter overlaps without an assumed closure of the adaptive recursion. These results yield exact regret curves for Thompson sampling, posterior-mean greedy selection, and a family of policies that scale the posterior sampling covariance. The normalized realized cumulative regret converges in L1, uniformly on compact proportional-time intervals. A policy-uniform lower bound identifies the limiting optimal Bayes regret and proves that posterior-mean greedy selection attains it. Thompson sampling incurs a strictly larger leading regret; its instantaneous regret ratio relative to greedy selection lies between one and two and approaches two at long proportional horizons. Closed-form cumulative curves also identify a different comparison in the vanishing-noise limit. Finally, the instantaneous regret converges to a nondegenerate Gaussian decision-loss distribution, rather than to its mean. The analysis separates the amount of information acquired by a bandit policy from the quality of the decisions made using that information.

The Baringhaus-Henze-Epps-Pulley (BHEP) tests for multivariate normality are affine-invariant goodness-of-fit tests based on a Gaussian-weighted $L^2$ distance between empirical and Gaussian characteristic functions. In 1990, Henze and Zirkler expressed the limiting null distribution through the eigenvalues of an integral operator on the standard Gaussian space. In 1997, Henze and Wagner obtained a simpler covariance kernel and raised the problem of calculating the eigenvalues of the resulting operator on a Gaussian-weighted space. Although subsequent work treated the univariate case and numerical approximations in a few low dimensions, the complete all-dimensional spectral problem remained open. This paper determines both complete spectra for every dimension $d \in \mathbb{N}$ and every smoothing parameter $\beta > 0$. The two operators are shown to have the forms $\mathcal{X}_{\beta,d}^*\mathcal{X}_{\beta,d}$ and $\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^*$ for the same Hilbert-Schmidt operator $\mathcal{X}_{\beta,d}$. Consequently, their nonzero eigenvalues agree, including multiplicities, while the null space of the Henze-Zirkler operator is identified exactly. The Gaussian integral operator in the Henze-Wagner decomposition is diagonalized by Mehler's formula, and rotational symmetry confines the finite-rank correction to the sectors associated with spherical harmonics of degrees $0$, $1$, and $2$. The degree-$1$ and degree-$2$ eigenvalues are characterized by scalar transcendental equations, and the radial eigenvalues by an explicit pole-safe Fredholm determinant. The paper establishes nonnegativity, multiplicities, eigenfunction reconstruction, completeness, the trace identity, and a complete characterization of all exceptional pole cases.

We introduce Sharper Transductive Local Complexity (STLC), a localized complexity method for transductive learning under uniform sampling without replacement. The construction starts from a Bernstein-type concentration inequality for the supremum of the test–train empirical process. Its proof uses the modified log-Sobolev inequality for the swap walk and a two-parameter entropy closure. A peeling argument with a surrogate localization functional then gives excess-risk bounds with the same fixed-point and confidence terms as the classical inductive local Rademacher-complexity bounds, without the additional logarithmic confidence factor in earlier transductive results. For realizable learning over a binary class of VC dimension $\dVC$, with training size $m$, test size $u$, and $u\ge m\ge\dVC$, STLC yields $\cO\{\dVC\log(me/\dVC)/m\}$. This matches the standard inductive rate and, when $m\ge9$, is within a logarithmic factor of the transductive minimax lower bound of order $\dVC/m$. For transductive kernel learning, STLC gives a spectrum-adaptive excess-risk bound without the multiplicative imbalance factors appearing in the earlier local-complexity bound.

We consider the construction of confidence intervals for population means when observations are subject to missingness. To accommodate more general missingness mechanisms than missing completely at random (MCAR), we adopt a reparametrised version of the realisable contamination model of Ma et al. (2026), which is a mixture of an MCAR version and a missing not at random version of the same base distribution $P$. We characterise the minimax length of confidence intervals that can adapt to potentially unknown parameters of the model, including the contamination fraction, for Gaussian base distributions and for nonparametric classes satisfying certain tail or symmetry assumptions. In all of these settings, we provide explicit constructions of simple, practical and finite-sample valid adaptive confidence intervals that attain the corresponding minimax rates. Finally, we provide implications of our results for causal inference.

We propose a class of extended generalized Pareto models for right-censored survival data, with particular emphasis on tail inference and long-term extrapolation. Our framework integrates extreme value theory and survival analysis, combining a generalized Pareto distribution with a flexible perturbation distribution on the unit interval. We consider three perturbation specifications: a parametric Beta model, a Bernstein polynomial estimator, and a histogram-based estimator. To facilitate direct comparison, all three models are fitted using a unified iterative procedure adapted to right censoring through Kaplan-Meier-based pseudo-observations. A Monte Carlo study evaluates finite-sample performance across different tail indices, censoring levels, sample sizes, and model complexities. The results reveal a trade-off between flexibility and stability: the Beta specification generally performs best for tail-index estimation, whereas the histogram estimator performs particularly well for scale estimation under low censoring. The Bernstein estimator shows intermediate performance and greater sensitivity to sample size and censoring. Applications to bladder cancer recurrence and heart-failure survival data show that models with very similar in-sample fits can nevertheless produce markedly different tail-index estimates and long-term extrapolations. These findings emphasize the importance of perturbation specification when extended generalized Pareto models are used for survival extrapolation under censoring.

We investigate $c$-interior pointwise maximal leakage (PML) as a tool for contraction analyses and disclosure control. Based on the strong adversarial threat models from maximal leakage, $c$-interior PML generalizes local differential privacy (LDP) to data-generating distributions with densities uniformly bounded away from zero by $c>0$. Viewing $c$-interior PML as an algebraic constraint on a kernel yields more flexible (and often tighter) contraction analyses than standard LDP. We provide tight bounds on the Dobrushin coefficient, and bound the contraction coefficient of the Hockeystick-divergence. We further derive strong data processing inequalities on $f$-divergences under $c$-interior PML constraints when the input distributions to the divergence are restricted to be in the $c$-interior. These results extend beyond the regime of pure LDP to cover a larger class of kernels, including, e.g., arbitrary stochastic matrices. We apply the results to minimax theory and provide asymptotically optimal strategies under $c$-interior PML constraints for binary hypothesis testing and mean estimation. The results show that disclosure control with PML allows analysts to reason about systems in a more differentiated manner: For example, it allows us to quantify the privacy leakage of deterministic systems, and can give precise adversarial guarantees with respect to arbitrary distributional assumptions. Interestingly, a recurring theme in the disclosure analyses is that if the privacy problem is relatively regular (if the density bound $c$ is large), private inference can be possible without incurring any additional cost in terms of sample complexity.

We study the nonparametric statistical inverse problem of recovering the space-dependent permittivity in a coupled Fokker-Planck-Darcy system from discrete, noisy observations of the Fokker-Planck solution. We consider the coupled parabolic-elliptic system on a bounded domain with the physically natural no-flux boundary condition for the Fokker-Planck equation and homogeneous Dirichlet boundary conditions for the Darcy equation. The inverse problem is indirect, since the unknown coefficient enters only through the elliptic equation and is observed only through its effect on the density in the parabolic equation. We first develop the analytical theory of the forward problem required for the statistical analysis, establishing well-posedness and a priori bounds uniform over the admissible set of permittivities. We then derive two stability estimates for the inverse problem, namely a Lipschitz-type forward estimate and a generalized backward estimate. Placing a rescaled Gaussian process prior on the log-shifted permittivity, we show that the posterior contracts around the truth at an explicit polynomial rate in the number of observations, with the posterior mean converging at the same rate. Numerical experiments in one and two dimensions, using preconditioned Crank-Nicolson and ensemble Kalman filter algorithms, complement our theoretical results.

We establish a Donsker-Prokhorov invariance principle in some Besov spaces. Specifically, we show that polygonal line processes associated with partial sum processes of a triangular array of row-wise independent random variables converge in distribution to Brownian motion, extending earlier results in the literature.

Blest's rank correlation $\nu$ is a variant of Spearman's rho $\rho$ that weights the leading ranks of one variable more heavily, at the price that $\nu$ is not symmetric in its arguments. We quantify both features by determining the exact region of $(\rho,\nu)$ over all bivariate copulas, as well as that of $(\eta,\nu)$, where $\eta$ is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs $(\nu(C),\nu(C^\top))$ formed by a copula $C$ and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most $1/4$, and interchanging the two variables changes it by at most $27/64$, improving on the bound $1/2$ implied by the first inequality. For every given value of $\rho$ or $\eta$, each corresponding extreme value of $\nu$ is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the $(\eta,\nu)$-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.

We study asymmetric simple binary hypothesis testing between $H_0:P_0^{n}$ and $H_1:P_1^{n}$, based on $n$ independent and identically distributed observations. Leveraging a variational representation of Rényi divergence of order $\alpha$, we derive our main result: a finite-sample converse with $\alpha>1$. The bound uses both directions of the divergence $D_\alpha(P_1\|P_0)$ and $D_\alpha(P_0\|P_1)$, tensorises under product measures, and contains familiar data-processing converses as boundary cases. For comparison, we apply the same variational approach to general $f$-divergences and specialise it to total variation, $E_\gamma$, Hellinger, and Kullback Leibler divergences, thereby recovering familiar converses within a unified framework. Together with an achievability bound involving Rényi divergence with $\alpha\in (0,1)$, the main converse recovers the phase transition of the optimal Type II error under the exponentially decaying Type I error constraint $\varepsilon_n=e^{-nr}$. Under regularity conditions, the optimal Type II error vanishes exponentially when $r<D(P_1\|P_0)$ and converges exponentially fast to one when $r>D(P_1\|P_0)$. We also derive sample-complexity bounds and extend both the converse and achievability analyses to locally differentially private observations, quantifying the cost of privacy and recovering the non-private achievability bound as the privacy constraint vanishes.

Score-based diffusion models are increasingly considered in settings where the underlying data distribution may differ from the training distribution, yet existing theoretical guarantees largely focus on the no-shift setting. In this work, we study robust score estimation under Wasserstein perturbations of a reference distribution. For the Ornstein–Uhlenbeck diffusion, we show that robust estimation decomposes into two fundamental components: the statistical cost of learning the reference distribution and the intrinsic cost of distribution shift. The latter scales quadratically with the Wasserstein radius, and this dependence is minimax optimal. We construct an explicit finite-sample estimator achieving the resulting robust minimax rate without knowing the shift radius. When the reference distribution lies on an unknown low-dimensional subspace, the statistical term adapts to the intrinsic dimension while the shift cost remains unchanged. Finally, we show that the same decomposition governs positive-time reverse sampling and obtain matching minimax guarantees in KL divergence. Together, these results characterize how finite data, intrinsic dimension, and distribution shift affect the robustness of score-based diffusion models.

We study sequential testing of a fixed family of hypotheses when observations are costly and a sampling horizon is specified in advance. The challenge is to pool evidence for earlier decisions when the number and identities of false hypotheses are unknown, while controlling the probability of any false rejection at level $\alpha$. Existing merges attain the pooled growth rate at a single number of false hypotheses: averaging when one is false, multiplying when all are. Under an independent-stream model with common simple null and alternative distributions, we test each intersection with a prior-weighted mixture of products of marginal likelihood ratios. Closed testing combines these elementary-symmetric-polynomial mixtures to identify individual false hypotheses. Design-specific boundary calibration gives finite-horizon family-wise error control, with exact finite-state guarantees or a confidence qualification for Monte Carlo calibration. The prior-matched mixture uniquely maximizes expected log evidence at each horizon. Mixtures assigning positive weight to every nonempty subset of streams attain log-growth rate $lD$ when $l$ streams follow the alternative. Here $D$ is the mean log likelihood ratio per alternative observation, and a round supplies one observation per stream. This rate attains the first-order intersection-delay lower bound as $\alpha\downarrow0$ at fixed dimension, configuration, weights, and a long enough horizon. Power for an individual hypothesis cannot exceed the best single-stream power at a given deadline, but closure removes the multiplicity penalty when all are false. Gaussian, basket-trial, language-model and advertising studies illustrate both. The primary basket boundaries are 40-52% below $1/\alpha$. Across 41 simulated configurations, the median reduction in capped mean patient outcomes relative to prespecified interim-look Bonferroni tests is 31%.

A latent severity has a bounded lower tail with density of shape alpha and scale L. It is observed only through a fixed Markov kernel K that is biased and non-additive. The relative conditional spread of K diverges at the endpoint. Our sample is i.i.d. from the marginal Q alone, with no anchoring covariate or instrument. We prove a dichotomy. The shape index alpha is identifiable: for every admissible choice of the class constants, any two observationally equivalent members of a lean class share alpha, determined by a near-endpoint expansion of Q. The rate, namely L and the fixed-scale exceedance p_tau, does not survive. There exist admissible shared class constants and two members of a smaller regularity class whose observed laws coincide exactly. Across the pair alpha agrees, whereas L and p_tau move. A degenerate Le Cam two-point bound excludes any uniformly consistent estimator of either, and pointwise consistency fails at one member. Only the rate needs an anchor. We conjecture that a known kernel family with known edge map identifies the rate fiber by fiber if and only if the family satisfies a fixed-scale injectivity clause, and we prove the sufficiency direction. In surrogate safety, uncalibrated conflict data give the shape of near-crash risk, not its absolute rate.

This work revisits a fundamental question in active learning: how powerful is the ability to synthesize arbitrary queries? Compared to pool-based active learning, where the learner only selects queries from a given unlabeled pool, we find that this seemingly mild change in query ability may dramatically alter the difficulty of statistical learning. In particular, some hypothesis classes that are inherently slow to learn in the pool-based setting, achieving only polynomial error decay in the number of samples, become exponentially learnable once synthesized queries are allowed. This striking gap suggests that membership query synthesis induces a fundamentally different mode of learning, one that is not adequately captured by existing active learning theory and calls for new analytical tools to characterize its complexity. Motivated by this phenomenon, we develop several sufficient conditions, present intriguing examples, and propose a conjectural perspective toward understanding which hypothesis classes admit efficient learning through synthesized queries.

We study identity testing for high-dimensional covariance matrices against dense alternatives of unknown direction, with $p/n \to \gamma$. Along a globally positive quadratic precision path, mixing Gaussian alternatives over a Gaussian Orthogonal Ensemble direction yields a contiguous experiment whose log likelihood reduces to the corrected Frobenius statistic; its upper-tail test attains the limiting weighted-power envelope at every fixed strength. Fixing the prior's Frobenius radius perturbs the mixture by only $O(p^{-1/2})$ in total variation, and exact whitening carries the experiment, the statistic, and its null law to any known null covariance. Separately, under a product-coordinate null, feasibility needs only $4+\eta$ moments, plus identical distributions over time when means are estimated; studentization and an exact degrees-of-freedom correction preserve the local power. A stability inequality turns near-envelope attainment into null agreement with the Frobenius rule, so uniform noninferiority on the typical dense bulk precludes gains at any contiguous alternative. For trace-matched rank-one alternatives, the corrected statistic is the first likelihood direction when $\vartheta_n \to 0$ and $n\vartheta_n \to \infty$; at fixed strength, the log likelihood ratio in the Onatski-Moreira-Hallin fixed-spike benchmark is governed by a richer linear spectral statistic below the Baik-Ben Arous-Peche threshold, while eigenvalue separation permits cost-free largest-eigenvalue enhancement above it. Simulations illustrate the theory.

We develop a general framework for model specification testing in stationary functional time series. The approach is based on an autoregressive approximation that represents a broad class of stationary functional processes through coefficient kernels whose dimension and autoregressive order may increase with the sample size. Different model assumptions induce different structural restrictions on these kernels, and our tests are constructed by measuring deviations from the corresponding restrictions. We illustrate this principle for three problems: testing a prescribed order of a functional autoregressive model, testing a functional autoregressive moving-average specification, and testing separability of autoregressive coefficient kernels. The resulting statistics are based on weighted $\mathcal{L}^2$-distances, and the critical values are obtained by a multiplier bootstrap. We establish a quantitative bootstrap approximation that is uniform over a class of weight functions and prove asymptotic validity and consistency of the proposed tests. The methodology allows for data-adaptive weighting and is illustrated by simulations and a data example.